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}{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 266 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Complete permutations" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 19.8.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The letterbox problem - simple cases " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 263 8 "Question" } {TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 166 "At the end of her rou nd a postie has 10 different letters left to deliver to10 different pe ople. As she is tired, she just stuffs them into the letterboxes at ra ndom." }}{PARA 0 "" 0 "" {TEXT -1 76 "Find the probabilty that none of the 10 people receive their correct letter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 8 "Solution" }{TEXT -1 3 ": " }} {PARA 0 "" 0 "" {TEXT -1 105 "The problem is straightforward to solve \+ if the number of letters involved is smaller, say 3 or 4 letters." }} {PARA 0 "" 0 "" {TEXT -1 45 "In these cases we can list the possibilit ies." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "W e start by considering " }{TEXT 259 9 "3 letters" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 125 "Given 3 letters (the upper case letter s ymbols rather than the sort that are posted) A, B, C initially in alph abetical order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 299 "You could imagine that there are 3 cards with one of the three letters A, B and C written on each card, and that the cards are initially placed in alphabetical order on a table. The new arrangemen t, could be achieved by lifting them, shuffling them, and then placing them back on the table in line. " }}{PARA 13 "" 1 "" {GLPLOT2D 376 113 113 {PLOTDATA 2 "6*-%)POLYGONSG6$7&7$$\"\"!F)F(7$$\"\"\"F)F(7$F+$ \"\"#F)7$F(F.-%&COLORG6&%$RGBG$\"#&*!\"#$\"#&)F7F,-F$6$7&7$F.F(7$$\"\" $F)F(7$F?F.7$F.F.-F26&F4F,F8F5-F$6$7&7$$\"\"%F)F(7$$\"\"&F)F(7$FLF.7$F IF.-F26&F4F5F,F8-%%TEXTG6%7$$FM!\"\"F+Q\"A6\"-F26&F4$\"#lF7F)$\"\")FW- FS6%7$$\"#DFWF+Q\"BFY-F26&F4FhnF)Ffn-FS6%7$$\"#XFWF+Q\"CFY-F26&F4FfnFh nF)-%%FONTG6%%&TIMESG%&ROMANG\"#[-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" }}}{PARA 0 "" 0 "" {TEXT -1 23 "We ca n easily list the " }{XPPEDIT 18 0 "3!=6" "6#/-%*factorialG6#\"\"$\"\" '" }{TEXT -1 155 " arrangements or permutations of these letters in a \+ line, and count the the number of these permutations for which no lett er occupies its initial position." }}{PARA 0 "" 0 "" {TEXT -1 31 "The \+ possible permutations are: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([[A,B,C],`all 3 letters in original position` ],[[A,C,B],`A in original position`],[[C,B,A],`B in original position` ],[[B,A,C],`C in original position`],[[B,C,A],`no letter in original p osition`],[[C,A,B],`no letter in original position`])" "6#-%*PIECEWISE G6(7$7%%\"AG%\"BG%\"CG%Call~3~letters~in~original~positionG7$7%F(F*F)% 7A~in~original~positionG7$7%F*F)F(%7B~in~original~positionG7$7%F)F(F*% 7C~in~original~positionG7$7%F)F*F(%?no~letter~in~original~positionG7$7 %F*F(F)F7" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }} {PARA 258 "" 0 "" {TEXT -1 77 "There are two permutations in which no \+ letter occupies its original location." }}{PARA 258 "" 0 "" {TEXT -1 164 "If we assume that the 6 possible permutations are equally likely, then the probabilty that, after rearranging the letters, no card occu pies its initial position is " }{XPPEDIT 18 0 "2/6=1/3" "6#/*&\"\"#\" \"\"\"\"'!\"\"*&F&F&\"\"$F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 28 "The auxiliary function L(k)." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 201 "I shall introduce a fu nction L(k) which gives the number of ways of re-arranging k objects, \+ such as k letters of the alphabet written on separate cards, so that n o letter occupies its initial position." }}{PARA 0 "" 0 "" {TEXT -1 106 "The probabilty of obtaining an permutation of this type when the \+ objects are re-arranged randomly is then " }{XPPEDIT 18 0 "L(k)/k!;" " 6#*&-%\"LG6#%\"kG\"\"\"-%*factorialG6#F'!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 29 "The example above shows that " }{XPPEDIT 18 0 "L(3) = 2;" "6#/-%\"LG6#\"\"$\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "It is easy to see that " }{XPPEDIT 18 0 "L(1) = 0;" "6#/-%\"LG6#\"\"\"\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "L(2)=1" " 6#/-%\"LG6#\"\"#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "To calculate L(4) we can procede as follows." }}{PARA 0 "" 0 "" {TEXT -1 72 "Consider 4 letters A, B, C, D initially arranged in alpha betical order. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 460 121 121 {PLOTDATA 2 "6,-%)POLYGONSG6$7&7$$\"\"!F)F(7$$\"\"\"F)F(7$F+$ \"\"#F)7$F(F.-%&COLORG6&%$RGBG$\"#&*!\"#$\"#&)F7F,-F$6$7&7$F.F(7$$\"\" $F)F(7$F?F.7$F.F.-F26&F4F,F8F5-F$6$7&7$$\"\"%F)F(7$$\"\"&F)F(7$FLF.7$F IF.-F26&F4F5F,F8-F$6$7&7$$\"\"'F)F(7$$\"\"(F)F(7$FYF.7$FVF.-F26&F4F,F5 F8-%%TEXTG6%7$$FM!\"\"F+Q\"A6\"-F26&F4$\"#lF7F)$\"\")F^o-Fjn6%7$$\"#DF ^oF+Q\"BF`o-F26&F4FeoF)Fco-Fjn6%7$$\"#XF^oF+Q\"CF`o-F26&F4FcoFeoF)-Fjn 6%7$$FdoF^oF+Q\"DF`o-F26&F4FeoFcoF)-%%FONTG6%%&TIMESG%&ROMANG\"#[-%*AX ESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "First consider the permutations with" }{TEXT 259 42 " exactly 1 le tter in its original position" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "There are " }{XPPEDIT 18 0 "C(4,1)=4" "6#/-%\"CG6$\"\"%\" \"\"F'" }{TEXT -1 106 " choices for this single letter. The remaining \+ 3 letters must be arranged in the remaining 3 positions in " } {XPPEDIT 18 0 "L(3)=2" "6#/-%\"LG6#\"\"$\"\"#" }{TEXT -1 58 " ways wit h no letter in its original position. This gives " }{XPPEDIT 18 0 "C(4 ,1)*`.`*L(3)=4*`.`*2 " "6#/*(-%\"CG6$\"\"%\"\"\"F)%\".GF)-%\"LG6#\"\"$ F)*(F(F)F*F)\"\"#F)" }{TEXT -1 4 " = " }{TEXT 259 14 "8 permutations " }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 50 "exactly 1 letter in its original position ----- " }{XPPEDIT 18 0 "PIECEWISE([[A, C, D, \+ B], `A in original position`],[[A, D, B, C], `A in original position`] ,[[C, B, D, A], `B in original position`],[[D, B, A, C], `B in origina l position`],[[B, D, C, A], `C in original position`],[[D, A, C, B], ` C in original position`],[[B, C, A, D], `D in original position`],[[C, A, B, D], `D in original position`])" "6#-%*PIECEWISEG6*7$7&%\"AG%\"C G%\"DG%\"BG%7A~in~original~positionG7$7&F(F*F+F)F,7$7&F)F+F*F(%7B~in~o riginal~positionG7$7&F*F+F(F)F17$7&F+F*F)F(%7C~in~original~positionG7$ 7&F*F(F)F+F67$7&F+F)F(F*%7D~in~original~positionG7$7&F)F(F+F*F;" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Now consider the permutations with " }{TEXT 259 44 "exact ly 2 letters in their original position" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 10 "There are " }{XPPEDIT 18 0 "C(4,2) = 6;" "6#/-%\"C G6$\"\"%\"\"#\"\"'" }{TEXT -1 106 " choices for this single letter. Th e remaining 2 letters must be arranged in the remaining 2 positions in " }{XPPEDIT 18 0 "L(2) = 1;" "6#/-%\"LG6#\"\"#\"\"\"" }{TEXT -1 57 " \+ way with no letter in its original position. This gives " }{XPPEDIT 18 0 "C(4,2)*`.`*L(2) = 6*`.`*1;" "6#/*(-%\"CG6$\"\"%\"\"#\"\"\"%\".GF *-%\"LG6#F)F**(\"\"'F*F+F*F*F*" }{XPPEDIT 18 0 "`` = 6;" "6#/%!G\"\"' " }{TEXT -1 4 " = " }{TEXT 259 14 "6 permutations" }{TEXT -1 1 "." }} {PARA 257 "" 0 "" {TEXT -1 48 " exactly 2 letters in original position ----- " }{XPPEDIT 18 0 "PIECEWISE([[A, B, D, C], `A and B in origin al positions`],[[A, D, C, B], `A and C in original positions`],[[A, C, B, D], `A and D in original positions`],[[D, B, C, A], `B and C in or iginal positions`],[[C, B, A, D], `B and D in original positions`],[[B , A, C, D], `C and D in original positions`])" "6#-%*PIECEWISEG6(7$7&% \"AG%\"BG%\"DG%\"CG%>A~and~B~in~original~positionsG7$7&F(F*F+F)%>A~and ~C~in~original~positionsG7$7&F(F+F)F*%>A~and~D~in~original~positionsG7 $7&F*F)F+F(%>B~and~C~in~original~positionsG7$7&F+F)F(F*%>B~and~D~in~or iginal~positionsG7$7&F)F(F+F*%>C~and~D~in~original~positionsG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "It is impossible to have exactly 3 letters in their original posi tions without the 4th and last letter also being in its original posit ion, so there is just 1 more permutation in which " }{TEXT 259 53 "at \+ least one letter does occupy its original position" }{TEXT -1 17 " for a total of " }{XPPEDIT 18 0 "C(4,1)*`.`*L(3)+C(4,2)*`.`*L(2)+1 = 15 " "6#/,(*(-%\"CG6$\"\"%\"\"\"F*%\".GF*-%\"LG6#\"\"$F*F**(-F'6$F)\"\"#F *F+F*-F-6#F3F*F*F*F*\"#:" }{TEXT -1 14 " permutations." }}{PARA 0 "" 0 "" {TEXT -1 80 "The number of permutations in which no letter occupi es its original position is " }{XPPEDIT 18 0 "4!-15;" "6#,&-%*factoria lG6#\"\"%\"\"\"\"#:!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= 24-15" " 6#/%!G,&\"#C\"\"\"\"#:!\"\"" }{TEXT -1 3 " = " }{TEXT 259 14 "9 permut ations" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " } {XPPEDIT 18 0 "L(4)=9" "6#/-%\"LG6#\"\"%\"\"*" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "An inductive formula fo r the number " }{XPPEDIT 18 0 "L(k)" "6#-%\"LG6#%\"kG" }{TEXT -1 29 " \+ of complete permutations of " }{TEXT 292 1 "k" }{TEXT -1 30 " objects \+ and letterbox problem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 124 "The examples of the previous section s uggest how we can construct an inductive formula for the number of re- arrangements of " }{TEXT 268 1 "k" }{TEXT -1 91 " objects so that no o bject occupies its original position. Such arrangements may be called \+ " }{TEXT 259 21 "complete permutations" }{TEXT -1 11 ". The term " } {TEXT 259 12 "derangements" }{TEXT -1 81 " is also sometimes used. Let the number of complete permutations of k objects be " }{XPPEDIT 18 0 "L(k);" "6#-%\"LG6#%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "We have seen that " }{XPPEDIT 18 0 "L(1) = 0,L(2) = 1,L(3) = 2;" " 6%/-%\"LG6#\"\"\"\"\"!/-F%6#\"\"#F'/-F%6#\"\"$F," }{TEXT -1 5 " and " }{XPPEDIT 18 0 "L(4) = 9;" "6#/-%\"LG6#\"\"%\"\"*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 42 "Now start to classify the permutations of " }{TEXT 267 1 "k" }{TEXT -1 16 " objects, where " }{XPPEDIT 18 0 "k \+ > 2" "6#2\"\"#%\"kG" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "matrix([[`no. of permutations with`], [`exactly 1 o bject in original position`]]);" "6#-%'matrixG6#7$7#%9no.~of~permutati ons~withG7#%Fexactly~1~object~in~original~positionG" }{TEXT -1 6 " = " }{XPPEDIT 18 0 "matrix([[`no. of ways of choosing`],[`1 object fro m k`])" "6#-%'matrixG6#7$7#%8no.~of~ways~of~choosingG7#%01~object~from ~kG" }{TEXT -1 8 " times " }{XPPEDIT 18 0 "matrix([[`no. of ways of a rranging`], [`the remaining (k-1) objects so that`], [` none occupies \+ its original position`]]);" "6#-%'matrixG6#7%7#%9no.~of~ways~of~arrang ingG7#%Dthe~remaining~(k-1)~objects~so~thatG7#%E~none~occupies~its~ori ginal~positionG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= C(k,1)*`.`*L(k-1) " "6#/%!G*(-%\"CG6$%\"kG\"\"\"F*%\".GF*-%\"LG6#,&F)F*F*!\"\"F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[`no. of permutati ons with`], [`exactly 2 objects in original position`]]);" "6#-%'matri xG6#7$7#%9no.~of~permutations~withG7#%Gexactly~2~objects~in~original~p ositionG" }{TEXT -1 6 " = " }{XPPEDIT 18 0 "matrix([[`no. of ways o f choosing`], [`2 objects from k`]]);" "6#-%'matrixG6#7$7#%8no.~of~way s~of~choosingG7#%12~objects~from~kG" }{TEXT -1 8 " times " }{XPPEDIT 18 0 "matrix([[`no. of ways of arranging`], [`the remaining (k-2) obje cts so that`], [` none occupies its original position`]]);" "6#-%'matr ixG6#7%7#%9no.~of~ways~of~arrangingG7#%Dthe~remaining~(k-2)~objects~so ~thatG7#%E~none~occupies~its~original~positionG" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = C(k,2)*`.`*L(k-2);" "6#/%!G*(-%\"CG6$%\"kG\"\"#\" \"\"%\".GF+-%\"LG6#,&F)F+F*!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 59 "We obtain a sequence of such formulas of the general fo rm: " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[`no . of permutations with`], [`exactly j objects in original position`]]) ;" "6#-%'matrixG6#7$7#%9no.~of~permutations~withG7#%Gexactly~j~objects ~in~original~positionG" }{TEXT -1 6 " = " }{XPPEDIT 18 0 "matrix([[ `no. of ways of choosing`], [`j objects from k`]]);" "6#-%'matrixG6#7$ 7#%8no.~of~ways~of~choosingG7#%1j~objects~from~kG" }{TEXT -1 8 " time s " }{XPPEDIT 18 0 "matrix([[`no. of ways of arranging`], [`the remain ing (k-j) objects so that`], [` none occupies its original position`]] );" "6#-%'matrixG6#7%7#%9no.~of~ways~of~arrangingG7#%Dthe~remaining~(k -j)~objects~so~thatG7#%E~none~occupies~its~original~positionG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = C(k,j)*`.`*L(k-j);" "6#/%!G*(-%\"CG6$%\" kG%\"jG\"\"\"%\".GF+-%\"LG6#,&F)F+F*!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "until we reach:" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[`no. of permutations with`], [`exactly (k-2) objects in original position`]]);" "6#-%'matrixG6#7$7#%9no.~of~ permutations~withG7#%Kexactly~(k-2)~objects~in~original~positionG" } {TEXT -1 6 " = " }{XPPEDIT 18 0 "matrix([[`no. of ways of choosing` ], [`(k-2) objects from k`]]);" "6#-%'matrixG6#7$7#%8no.~of~ways~of~ch oosingG7#%5(k-2)~objects~from~kG" }{TEXT -1 8 " times " }{XPPEDIT 18 0 "matrix([[`no. of ways of arranging`], [`the remaining 2 objects so \+ that`], [` none occupies its original position`]]);" "6#-%'matrixG6#7% 7#%9no.~of~ways~of~arrangingG7#%@the~remaining~2~objects~so~thatG7#%E~ none~occupies~its~original~positionG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = C(k,k-2)*`.`*L(2);" "6#/%!G*(-%\"CG6$%\"kG,&F)\"\"\"\"\"#!\"\"F+%\" .GF+-%\"LG6#F,F+" }{XPPEDIT 18 0 "`` = C(k,k-2)*`.`*1" "6#/%!G*(-%\"CG 6$%\"kG,&F)\"\"\"\"\"#!\"\"F+%\".GF+F+F+" }{XPPEDIT 18 0 "``=C(k,k-2) " "6#/%!G-%\"CG6$%\"kG,&F(\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 124 "We obtain just one more permutation in which at \+ least 1 object does not move, namely the permutation in which nothing \+ moves." }}{PARA 0 "" 0 "" {TEXT -1 79 "The total number of permutation s in which at least one object does not move is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C(k,1 )*`.`*L(k-1)+C(k,2)*`.`*L(k-2)+` . . . `+C(k,k-2)*`.`*L(2)+1;" "6#,,*( -%\"CG6$%\"kG\"\"\"F)%\".GF)-%\"LG6#,&F(F)F)!\"\"F)F)*(-F&6$F(\"\"#F)F *F)-F,6#,&F(F)F3F/F)F)%(~.~.~.~GF)*(-F&6$F(,&F(F)F3F/F)F*F)-F,6#F3F)F) F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Sum(C(k,j)*`.`*L(j),j=1..k-2)+ 1" "6#/%!G,&-%$SumG6$*(-%\"CG6$%\"kG%\"jG\"\"\"%\".GF/-%\"LG6#F.F//F.; F/,&F-F/\"\"#!\"\"F/F/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L(k) = n !-Sum(C(k,j)*`.`*L(j),j = 1 .. k-2)-1;" "6#/-%\"LG6#%\"kG,(-%*factoria lG6#%\"nG\"\"\"-%$SumG6$*(-%\"CG6$F'%\"jGF-%\".GF--F%6#F5F-/F5;F-,&F'F -\"\"#!\"\"F=F-F=" }{TEXT -1 14 " ------- (i)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 269 20 "____________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The proba bility of obtaining a complete permutation of k objects from a random \+ permutation is " }{XPPEDIT 18 0 "L(k)/n!" "6#*&-%\"LG6#%\"kG\"\"\"-%*f actorialG6#%\"nG!\"\"" }{TEXT -1 56 ", and so it can be easily calcula ted once L(k) is known." }}{PARA 0 "" 0 "" {TEXT -1 75 "In particular, the answer to the question given in the previous section is " } {XPPEDIT 18 0 "L(10)/10!" "6#*&-%\"LG6#\"#5\"\"\"-%*factorialG6#F'!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "The following Maple procedure implements the formula (i) \+ to define " }{XPPEDIT 18 0 "L(k)" "6#-%\"LG6#%\"kG" }{TEXT -1 13 " rec ursively." }}{PARA 0 "" 0 "" {TEXT -1 97 "It uses the \"remember\"opti on so that previously computed values do not have to be re-calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "L := proc(k::posint)\n local i;\n option remembe r;\n if k=1 then 0 elif k=2 then 1 else\n k!-add(binomial(k,i)* L(k-i),i=1..k-2)-1\n end if;\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6#'%\"kG%'posintG6#%\"iG6#%)rememberG6\"@'/9$\" \"\"\"\"!/F1\"\"#F2,(-%*factorialG6#F1F2-%$addG6$*&-%)binomialG6$F18$F 2-F$6#,&F1F2FA!\"\"F2/FA;F2,&F1F2F5FEFEF2FEF.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We can now obtain a list \+ of values for " }{XPPEDIT 18 0 "L(k)" "6#-%\"LG6#%\"kG" }{TEXT -1 38 " , and the corresponding probabilities " }{XPPEDIT 18 0 "p(k) = L(k)/k! ;" "6#/-%\"pG6#%\"kG*&-%\"LG6#F'\"\"\"-%*factorialG6#F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "printf(\"\\n k k! \+ L(k) p(k)\\n\\n\");\nfor k from 1 to 20 do\n t := k!;\n s := L(k);\n p := evalf[20](s/t); \n printf(\"%4d %23d%23d%2 5.16f\\n\",k,t,s,p);\nend do:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 69 " k k! \+ L(k) p(k)" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 76 " 1 1 \+ 0 0.0000000000000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 2 \+ 2 1 0.5000000000000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 3 6 \+ 2 0.3333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 76 " 4 24 9 0.3750000000 000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 5 120 \+ 44 0.3666666666666667" }}{PARA 6 "" 1 "" {TEXT -1 76 " 6 720 265 \+ 0.3680555555555556" }}{PARA 6 "" 1 "" {TEXT -1 76 " 7 \+ 5040 1854 0.3678571428571429" }}{PARA 6 "" 1 "" {TEXT -1 76 " 8 40320 148 33 0.3678819444444444" }}{PARA 6 "" 1 "" {TEXT -1 76 " 9 \+ 362880 133496 0.3678791887125220" }} {PARA 6 "" 1 "" {TEXT -1 76 " 10 3628800 \+ 1334961 0.3678794642857143" }}{PARA 6 "" 1 "" {TEXT -1 76 " \+ 11 39916800 14684570 0.367879439233 6059" }}{PARA 6 "" 1 "" {TEXT -1 76 " 12 479001600 \+ 176214841 0.3678794413212816" }}{PARA 6 "" 1 "" {TEXT -1 76 " 13 6227020800 2290792932 0.367 8794411606912" }}{PARA 6 "" 1 "" {TEXT -1 76 " 14 8717829 1200 32071101049 0.3678794411721619" }}{PARA 6 "" 1 " " {TEXT -1 76 " 15 1307674368000 481066515734 \+ 0.3678794411713972" }}{PARA 6 "" 1 "" {TEXT -1 76 " 16 20 922789888000 7697064251745 0.3678794411714450" }}{PARA 6 "" 1 "" {TEXT -1 76 " 17 355687428096000 13085009227 9664 0.3678794411714422" }}{PARA 6 "" 1 "" {TEXT -1 76 " 18 \+ 6402373705728000 2355301661033953 0.3678794411714423" }}{PARA 6 "" 1 "" {TEXT -1 76 " 19 121645100408832000 44750 731559645106 0.3678794411714423" }}{PARA 6 "" 1 "" {TEXT -1 76 " 20 2432902008176640000 895014631192902121 0.3678794411 714423" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "The answer to the question of the previous section is that the pr obability of none of the10 people receiving the correct letter is " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L(10)/10!=1334961/362 8800" "6#/*&-%\"LG6#\"#5\"\"\"-%*factorialG6#F(!\"\"*&\"(h\\L\"F)\"(+) GOF-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= 16481/44800" "6#/%!G*&\"&\"[ ;\"\"\"\"&+[%!\"\"" }{TEXT -1 2 " " }{TEXT 270 1 "~" }{TEXT -1 15 " 0 .3678794643. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "A sim pler inductive formula for " }{XPPEDIT 18 0 "L(k)" "6#-%\"LG6#%\"kG" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 108 "Consider the complete permutations of the 5 le tters A, B, C, D, E, initially arranged in alphabetical order." }} {PARA 0 "" 0 "" {TEXT -1 71 "There are 4 possible choices for the firs t letter namely, B, C, D or E." }}{PARA 0 "" 0 "" {TEXT -1 60 "Suppose , for example, that we choose C for the first letter." }}{PARA 0 "" 0 "" {TEXT -1 307 "We now investigate the possible permutations of the r emaining letters A, B, D and E in the positions of the original letter s B, C, D, E. Temporarily rename or replace the letter C by A among th e letters B, C, D, E of the last 4 letters in their original positions to form the configuration B, A, D, E. The " }{XPPEDIT 18 0 "L(4)=9" "6#/-%\"LG6#\"\"%\"\"*" }{TEXT -1 123 " complete permutations of the c onfiguration B, A, D, E give rise to 9 complete permutations of the 5 \+ letters A, B, C, D, E." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[`first letter C, followed by`], [`complete permutation s of B, A, D, E`]]);" "6#-%'matrixG6#7$7#%=first~letter~C,~~followed~b yG7#%Dcomplete~permutations~of~B,~A,~D,~EG" }{TEXT -1 10 " ----- " }{XPPEDIT 18 0 "PIECEWISE([[C, A, B, E, D], `A in 2nd position`],[[C, \+ A, D, E, B], `A in 2nd position`],[[C, A, E, B, D], `A in 2nd position `],[[C, D, B, E, A], `D in 2nd position`],[[C, D, E, B, A], `D in 2nd \+ position`],[[C, D, E, A, B], `D in 2nd position`],[[C, E, B, A, D], `E in 2nd position`],[[C, E, D, A, B], `E in 2nd position`],[[C, E, D, B , A], `E in 2nd position`]);" "6#-%*PIECEWISEG6+7$7'%\"CG%\"AG%\"BG%\" EG%\"DG%2A~in~2nd~positionG7$7'F(F)F,F+F*F-7$7'F(F)F+F*F,F-7$7'F(F,F*F +F)%2D~in~2nd~positionG7$7'F(F,F+F*F)F47$7'F(F,F+F)F*F47$7'F(F+F*F)F,% 2E~in~2nd~positionG7$7'F(F+F,F)F*F;7$7'F(F+F,F*F)F;" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "In thes e permutations A does not appear in the 3rd position." }}{PARA 0 "" 0 "" {TEXT -1 10 "There are " }{XPPEDIT 18 0 "L(3)=2" "6#/-%\"LG6#\"\"$ \"\"#" }{TEXT -1 47 " more permutations with A in the 3rd position. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[`first letter C and 3rd letter A, with`], [`co mplete permutations of B, D, E`], [`in remaining positions`]]);" "6#-% 'matrixG6#7%7#%Ffirst~letter~C~and~3rd~letter~A,~withG7#%Acomplete~per mutations~of~B,~D,~EG7#%7in~remaining~positionsG" }{TEXT -1 10 " --- -- " }{XPPEDIT 18 0 "PIECEWISE([[C, D, A, E, B], ``],[[C, E, A, B, D] , ``]);" "6#-%*PIECEWISEG6$7$7'%\"CG%\"DG%\"AG%\"EG%\"BG%!G7$7'F(F+F*F ,F)F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "This gives a total of " }{XPPEDIT 18 0 "L(4)+L(3) = \+ 11" "6#/,&-%\"LG6#\"\"%\"\"\"-F&6#\"\"$F)\"#6" }{TEXT -1 69 " complete permutations of A, B, C, D, E with C in the first position." }}{PARA 0 "" 0 "" {TEXT -1 21 "Similarly, there are " }{XPPEDIT 18 0 "L(4)+L(3 )" "6#,&-%\"LG6#\"\"%\"\"\"-F%6#\"\"$F(" }{TEXT -1 94 " complete permu tations with the other 3 choices for the first letter, which gives a t otal of " }{XPPEDIT 18 0 "4*`.`*(L(4)+L(3)) = 44;" "6#/*(\"\"%\"\"\"% \".GF&,&-%\"LG6#F%F&-F*6#\"\"$F&F&\"#W" }{TEXT -1 46 " complete permut ations of 5 letters, that is, " }{XPPEDIT 18 0 "L(5)=44" "6#/-%\"LG6# \"\"&\"#W" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "The precedin g argument generalises to give the inductive formula: " }}{PARA 257 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L(k)=(k-1)*(L(k-1)+L(k-2))" "6# /-%\"LG6#%\"kG*&,&F'\"\"\"F*!\"\"F*,&-F%6#,&F'F*F*F+F*-F%6#,&F'F*\"\"# F+F*F*" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 271 15 "_______________" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 5 " for " }{XPPEDIT 18 0 "k >=3" "6#1\"\"$%\"kG" }{TEXT -1 37 ". 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 85 ": This inductive formula is reminiscent of the inductive \+ formula which generates the " }{TEXT 259 17 "Fibonacci numbers" } {TEXT -1 1 " " }{XPPEDIT 18 0 "0,1,1,2,3,5,8,13,21,34,55,` . . . `;" " 6.\"\"!\"\"\"F$\"\"#\"\"$\"\"&\"\")\"#8\"#@\"#M\"#b%(~.~.~.~G" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([F(0)=0,``],[F(1)=1,``],[F(k)=F(k-1 )+F(k-2),k>=3])" "6#-%*PIECEWISEG6%7$/-%\"FG6#\"\"!F+%!G7$/-F)6#\"\"\" F1F,7$/-F)6#%\"kG,&-F)6#,&F6F1F1!\"\"F1-F)6#,&F6F1\"\"#F;F11\"\"$F6" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "See: " }{HYPERLNK 17 "co mbinat,fibonacci" 2 "combinat,fibonacci" "" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The following M aple procedure implements the formulas " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([L( 0) = 0, ``],[L(1) = 1, ``],[L(k) = (k-1)*(L(k-1)+L(k-2)), 3 <= k]);" " 6#-%*PIECEWISEG6%7$/-%\"LG6#\"\"!F+%!G7$/-F)6#\"\"\"F1F,7$/-F)6#%\"kG* &,&F6F1F1!\"\"F1,&-F)6#,&F6F1F1F9F1-F)6#,&F6F1\"\"#F9F1F11\"\"$F6" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 11 " to define " }{XPPEDIT 18 0 "L(k)" "6#-%\"LG6#%\"kG" }{TEXT -1 13 " recursively." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "L := proc(k::posint)\n local i;\n option remember;\n if k=1 then 0 e lif k=2 then 1 else\n (k-1)*(L(k-1)+L(k-2));\n end if;\nend pro c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6#'%\"kG%'posintG6#%\"iG 6#%)rememberG6\"@'/9$\"\"\"\"\"!/F1\"\"#F2*&,&F1F2F2!\"\"F2,&-F$6#F7F2 -F$6#,&F1F2F5F8F2F2F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can obtain the same results as in the previous section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "printf(\"\\n k k! \+ L(k) p(k)\\n\\n\");\nfor k from 1 to 20 do\n t := k!;\n s := L(k);\n p := evalf[20](s/t); \n printf(\"%4d %23d%23d%2 5.16f\\n\",k,t,s,p);\nend do:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 69 " k k! \+ L(k) p(k)" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 76 " 1 1 \+ 0 0.0000000000000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 2 \+ 2 1 0.5000000000000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 3 6 \+ 2 0.3333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 76 " 4 24 9 0.3750000000 000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 5 120 \+ 44 0.3666666666666667" }}{PARA 6 "" 1 "" {TEXT -1 76 " 6 720 265 \+ 0.3680555555555556" }}{PARA 6 "" 1 "" {TEXT -1 76 " 7 \+ 5040 1854 0.3678571428571429" }}{PARA 6 "" 1 "" {TEXT -1 76 " 8 40320 148 33 0.3678819444444444" }}{PARA 6 "" 1 "" {TEXT -1 76 " 9 \+ 362880 133496 0.3678791887125220" }} {PARA 6 "" 1 "" {TEXT -1 76 " 10 3628800 \+ 1334961 0.3678794642857143" }}{PARA 6 "" 1 "" {TEXT -1 76 " \+ 11 39916800 14684570 0.367879439233 6059" }}{PARA 6 "" 1 "" {TEXT -1 76 " 12 479001600 \+ 176214841 0.3678794413212816" }}{PARA 6 "" 1 "" {TEXT -1 76 " 13 6227020800 2290792932 0.367 8794411606912" }}{PARA 6 "" 1 "" {TEXT -1 76 " 14 8717829 1200 32071101049 0.3678794411721619" }}{PARA 6 "" 1 " " {TEXT -1 76 " 15 1307674368000 481066515734 \+ 0.3678794411713972" }}{PARA 6 "" 1 "" {TEXT -1 76 " 16 20 922789888000 7697064251745 0.3678794411714450" }}{PARA 6 "" 1 "" {TEXT -1 76 " 17 355687428096000 13085009227 9664 0.3678794411714422" }}{PARA 6 "" 1 "" {TEXT -1 76 " 18 \+ 6402373705728000 2355301661033953 0.3678794411714423" }}{PARA 6 "" 1 "" {TEXT -1 76 " 19 121645100408832000 44750 731559645106 0.3678794411714423" }}{PARA 6 "" 1 "" {TEXT -1 76 " 20 2432902008176640000 895014631192902121 0.3678794411 714423" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 83 "The limiti ng value for the probability of a complete permutation and a formula f or " }{XPPEDIT 18 0 "L(k)" "6#-%\"LG6#%\"kG" }{TEXT -1 2 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 170 "Acknowledgement: The following argument is suggested by an exe rcise on page129 in \"Introduction to Finite Mathematics\" by Kemeny, \+ Snell and Thompson, Prentice Hall, 1964." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "p(k)" "6#-%\" pG6#%\"kG" }{TEXT -1 48 " be the probabilty that a random permutation \+ of " }{TEXT 288 1 "k" }{TEXT -1 60 " letters is a complete permutation . The tabulated values of " }{XPPEDIT 18 0 "p(k)" "6#-%\"pG6#%\"kG" } {TEXT -1 28 " appear to approach a limit." }}{PARA 0 "" 0 "" {TEXT -1 19 "We shall show that " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Limit(p(k),k = infinity) = 1/exp(1);" "6#/-%&LimitG6$-% \"pG6#%\"kG/F*%)infinityG*&\"\"\"F.-%$expG6#F.!\"\"" }{TEXT -1 1 " " } {TEXT 272 1 "~" }{TEXT -1 21 " 0.3678794411714423. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 273 19 "___________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " } {XPPEDIT 18 0 "p(k)=L(k)/n!" "6#/-%\"pG6#%\"kG*&-%\"LG6#F'\"\"\"-%*fac torialG6#%\"nG!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "L(k)" "6#- %\"LG6#%\"kG" }{TEXT -1 43 " is the number of complete permutations of " }{TEXT 291 1 "k" }{TEXT -1 36 " letters. Furthermore, the function \+ " }{XPPEDIT 18 0 "L(k)" "6#-%\"LG6#%\"kG" }{TEXT -1 32 " can be define d inductively by: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([L(1) = 0, ``],[L(2) = 1, ``],[L(k) = (k-1)*(L(k-1)+L(k-2)) , 3 <= k]);" "6#-%*PIECEWISEG6%7$/-%\"LG6#\"\"\"\"\"!%!G7$/-F)6#\"\"#F +F-7$/-F)6#%\"kG*&,&F7F+F+!\"\"F+,&-F)6#,&F7F+F+F:F+-F)6#,&F7F+F2F:F+F +1\"\"$F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "It follows \+ that " }{XPPEDIT 18 0 "p(1)=0, p(2)=1/2" "6$/-%\"pG6#\"\"\"\"\"!/-F%6 #\"\"#*&F'F'F,!\"\"" }{TEXT -1 9 " and for " }{TEXT 290 1 "k" }{TEXT -1 1 " " }{TEXT 289 1 ">" }{TEXT -1 5 " 3. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(k)=L(k)/k!" "6#/-%\"pG6#%\"kG*&-%\"LG 6#F'\"\"\"-%*factorialG6#F'!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(k -1)*(L(k-1)+L(k-2))/k!" "6#*(,&%\"kG\"\"\"F&!\"\"F&,&-%\"LG6#,&F%F&F&F 'F&-F*6#,&F%F&\"\"#F'F&F&-%*factorialG6#F%F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(k-1)/k" "6#/%!G*&,&%\"kG\"\"\"F(!\"\"F(F'F)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "[L(k-1)/(k-1)!+L(k-2)/((k-1)*(k-2)!)]" "6#7#,&*&-%\" LG6#,&%\"kG\"\"\"F+!\"\"F+-%*factorialG6#,&F*F+F+F,F,F+*&-F'6#,&F*F+\" \"#F,F+*&,&F*F+F+F,F+-F.6#,&F*F+F5F,F+F,F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(k-1)/k" "6#/%!G*&,&%\"kG\"\"\"F(!\"\"F(F'F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[p(k-1)+p(k-2)/(k-1)]" "6#7#,&-%\"pG6#,&%\"kG\"\"\"F*! \"\"F**&-F&6#,&F)F*\"\"#F+F*,&F)F*F*F+F+F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(k-1)*p(k-1)/k+p(k-2)/k" "6#/%!G,&*(,&%\"kG\"\"\"F)!\"\"F)-%\"pG 6#,&F(F)F)F*F)F(F*F)*&-F,6#,&F(F)\"\"#F*F)F(F*F)" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "p(k)" "6#-%\"pG6#% \"kG" }{TEXT -1 26 " is given inductively by: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([p(1)=0 , ``],[p(2)=1/2 ,`` ] ,[p(k)=(k-1)*p(k-1)/k+p(k-2)/k , k>=3])" "6#-%*PIECEWISEG6%7$/-%\"pG6# \"\"\"\"\"!%!G7$/-F)6#\"\"#*&F+F+F2!\"\"F-7$/-F)6#%\"kG,&*(,&F9F+F+F4F +-F)6#,&F9F+F+F4F+F9F4F+*&-F)6#,&F9F+F2F4F+F9F4F+1\"\"$F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "Now let " }{XPPEDIT 18 0 "v(k)=p( k)-p(k-1)" "6#/-%\"vG6#%\"kG,&-%\"pG6#F'\"\"\"-F*6#,&F'F,F,!\"\"F0" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "We have " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(k)=(k-1)*p(k-1)/k+p(k-2)/k" "6 #/-%\"pG6#%\"kG,&*(,&F'\"\"\"F+!\"\"F+-F%6#,&F'F+F+F,F+F'F,F+*&-F%6#,& F'F+\"\"#F,F+F'F,F+" }{TEXT -1 3 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = p(k-1)+(-p(k-1)+p(k-2))/k;" "6#/%!G,&-%\"pG6# ,&%\"kG\"\"\"F+!\"\"F+*&,&-F'6#,&F*F+F+F,F,-F'6#,&F*F+\"\"#F,F+F+F*F,F +" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(k)-p(k-1)=-(p(k-1)-p(k-2))/k" "6 #/,&-%\"pG6#%\"kG\"\"\"-F&6#,&F(F)F)!\"\"F-,$*&,&-F&6#,&F(F)F)F-F)-F&6 #,&F(F)\"\"#F-F-F)F(F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Thus we have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " v(k) = -v(k-1)/k;" "6#/-%\"vG6#%\"kG,$*&-F%6#,&F'\"\"\"F-!\"\"F-F'F.F. " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "k>=3" "6#1\"\"$%\"kG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 " Since " }{XPPEDIT 18 0 "v(2)=1/2" "6#/-%\"vG6#\"\"#*&\"\"\"F)F'!\"\"" }{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "v(3)=-1/(2*`.`*3), v (4)=1/(2*`.`*3*`.`*4), ` . . . `, v(k)=(-1)^k/k!" "6&/-%\"vG6#\"\"$,$* &\"\"\"F**(\"\"#F*%\".GF*F'F*!\"\"F./-F%6#\"\"%*&F*F**,F,F*F-F*F'F*F-F *F2F*F.%(~.~.~.~G/-F%6#%\"kG*&),$F*F.F9F*-%*factorialG6#F9F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "This now gives the formula " } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(k)=1/2!-1/3!+` . . . `+(-1)^k/k!" "6#/-%\"pG6#%\"kG,**&\"\"\"F*-%*factorialG6#\"\"#!\"\" F**&F*F*-F,6#\"\"$F/F/%(~.~.~.~GF**&),$F*F/F'F*-F,6#F'F/F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum((-1)^i/i!,i = 2 .. k);" "6#/%!G-%$SumG6 $*&),$\"\"\"!\"\"%\"iGF+-%*factorialG6#F-F,/F-;\"\"#%\"kG" }{TEXT -1 16 " ------- (ii), " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 274 20 "____________________" }{TEXT -1 12 " " }}{PARA 0 "" 0 " " {TEXT -1 4 "for " }{XPPEDIT 18 0 "k>=2" "6#1\"\"#%\"kG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "exp(x)" "6#-% $expG6#%\"xG" }{TEXT -1 47 " can be expanded using its Maclaurin serie s as " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)=1+x+x ^2/2!+x^3/3!+x^4/4!+` . . . `+x^k/k!+` . . . `" "6#/-%$expG6#%\"xG,2\" \"\"F)F'F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F)*&F'\" \"%-F-6#F5F/F)%(~.~.~.~GF)*&)F'%\"kGF)-F-6#F;F/F)F8F)" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(p(k),k=infinity)=Sum((-1)^k/k!,k=2..infini ty)" "6#/-%&LimitG6$-%\"pG6#%\"kG/F*%)infinityG-%$SumG6$*&),$\"\"\"!\" \"F*F3-%*factorialG6#F*F4/F*;\"\"#F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "exp(-1) = 1/exp(1);" "6#/-%$expG6#,$\"\"\"!\"\"*&F(F(-F%6#F(F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 80 ": It follows from (ii) that the numbe r of complete permutations of k objects is " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "L(k) = k!*(1-1/1!+1/2!-1/3!+` . . . `+(-1)^k/k!);" "6#/ -%\"LG6#%\"kG*&-%*factorialG6#F'\"\"\",.F,F,*&F,F,-F*6#F,!\"\"F1*&F,F, -F*6#\"\"#F1F,*&F,F,-F*6#\"\"$F1F1%(~.~.~.~GF,*&),$F,F1F'F,-F*6#F'F1F, F," }{XPPEDIT 18 0 "``= k!*Sum((-1)^i/i!,i = 0 .. k)" "6#/%!G*&-%*fact orialG6#%\"kG\"\"\"-%$SumG6$*&),$F*!\"\"%\"iGF*-F'6#F2F1/F2;\"\"!F)F* " }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 275 26 "_ _________________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "This formula can be used to con struct the same table as before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "printf(\"\\n k \+ k! L(k) p(k)\\n\\n\");\nfor \+ k from 1 to 20 do\n s := add((-1)^j*1/j!,j=2..k);\n p := evalf[20](s );\n t := k!;\n s := s*t; \n printf(\"%4d %23d%23d%25.16f\\n\",k,t, s,p);\nend do:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 69 " k k! L(k) \+ p(k)" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 76 " 1 1 0 0.000 0000000000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 2 \+ 2 1 0.5000000000000000" }}{PARA 6 "" 1 " " {TEXT -1 76 " 3 6 2 \+ 0.3333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 76 " 4 \+ 24 9 0.3750000000000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 5 120 \+ 44 0.3666666666666667" }}{PARA 6 "" 1 "" {TEXT -1 76 " 6 \+ 720 265 0.3680555555555556" }}{PARA 6 "" 1 "" {TEXT -1 76 " 7 5040 \+ 1854 0.3678571428571429" }}{PARA 6 "" 1 "" {TEXT -1 76 " 8 40320 14833 0.3678819444 444444" }}{PARA 6 "" 1 "" {TEXT -1 76 " 9 362880 \+ 133496 0.3678791887125220" }}{PARA 6 "" 1 "" {TEXT -1 76 " 10 3628800 1334961 \+ 0.3678794642857143" }}{PARA 6 "" 1 "" {TEXT -1 76 " 11 \+ 39916800 14684570 0.3678794392336059" }}{PARA 6 "" 1 "" {TEXT -1 76 " 12 479001600 1762148 41 0.3678794413212816" }}{PARA 6 "" 1 "" {TEXT -1 76 " 13 \+ 6227020800 2290792932 0.3678794411606912" }} {PARA 6 "" 1 "" {TEXT -1 76 " 14 87178291200 3 2071101049 0.3678794411721619" }}{PARA 6 "" 1 "" {TEXT -1 76 " \+ 15 1307674368000 481066515734 0.367879441171 3972" }}{PARA 6 "" 1 "" {TEXT -1 76 " 16 20922789888000 \+ 7697064251745 0.3678794411714450" }}{PARA 6 "" 1 "" {TEXT -1 76 " 17 355687428096000 130850092279664 0.367 8794411714422" }}{PARA 6 "" 1 "" {TEXT -1 76 " 18 640237370572 8000 2355301661033953 0.3678794411714423" }}{PARA 6 "" 1 " " {TEXT -1 76 " 19 121645100408832000 44750731559645106 \+ 0.3678794411714423" }}{PARA 6 "" 1 "" {TEXT -1 76 " 20 2432902 008176640000 895014631192902121 0.3678794411714423" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "Simulations for the exp erimental probability of complete permutations " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 2 "; " }}}{PARA 0 "" 0 "" {TEXT -1 164 "In this se ction set up procedures which can be used to perform simulation experi ments to determine experimental probabilities associated with complete permutations." }}{PARA 0 "" 0 "" {TEXT -1 20 "The first procedure " } {TEXT 0 11 "randarrange" }{TEXT -1 58 " constructs random permutations of the members of a list. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "randarrange := proc(A::list)\nloca l AA,t,i,k,rdm;\n AA := A;\n k := nops(AA);\n for i to k do \n \+ rdm := irem(rand(),k-i+1)+i;\n #rdm := trunc(rand()*(k-i+1)*1 e-12)+i;\n t := AA[i];\n AA[i] := AA[rdm];\n AA[rdm] := t\n end do;\n AA\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% ,randarrangeGf*6#'%\"AG%%listG6'%#AAG%\"tG%\"iG%\"kG%$rdmG6\"F0C&>8$9$ >8'-%%nopsG6#F3?(8&\"\"\"F8(,&-%%iremG6$-%%randGF0,(F6F8%&F36#F;>FK&F36#F@>FNFJF3F0F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "randomize(): \nfor i to 6 do randarrange([A,B,C,D,E]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"DG%\"BG%\"EG%\"CG%\"AG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"BG%\"AG%\"DG%\"CG%\"EG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"EG%\"BG%\"DG%\"AG%\"CG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"AG%\"BG%\"CG%\"DG%\"EG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"DG%\"CG%\"AG%\"BG%\"EG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"BG%\"CG%\"EG%\"AG%\"DG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Alternatively, the combinatoric s package " }{TEXT 0 8 "combinat" }{TEXT -1 22 " contains a procedure \+ " }{TEXT 0 8 "randperm" }{TEXT -1 19 " which can be used." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "with( combinat,randperm):\nrandomize():\nfor i to 6 do randperm([A,B,C,D,E]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"DG%\"BG%\"CG%\"EG%\"A G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"CG%\"DG%\"BG%\"AG%\"EG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"CG%\"BG%\"EG%\"DG%\"AG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"DG%\"EG%\"BG%\"AG%\"CG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"CG%\"DG%\"AG%\"EG%\"BG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"DG%\"BG%\"EG%\"AG%\"CG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 19 "The next procedure " }{TEXT 0 11 "trial_perms" }{TEXT -1 28 " makes use of the procedure " }{TEXT 0 11 "randarrange" }{TEXT -1 112 " to provide simulated trials of an experiment involving permut ations of a specified number of different objects." }}{PARA 0 "" 0 "" {TEXT -1 348 "For each trial permutation the number of objects which h ave remained static is counted. At the end of the experiment the distr ibution is given as a list in which the 1st member of the list is the \+ number of trials where the permutation had exactly 1 static object, th e 2nd member of the list is the number of permutations with 2 static o bjects, etc." }}{PARA 0 "" 0 "" {TEXT -1 141 "The number of trials wit h no static objects is obtained by adding the members of the list and \+ subtracting this sum from the number of trials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 829 "trial_perms := proc(numobjects::posint,numtrials::posint)\n local objects,DD,i, j,c,shuff,randarrange;\n\n randarrange := proc(A::list)\n local AA ,t,i,k,rdm;\n AA := A;\n k := nops(AA);\n for i to k do \n rdm := irem(rand(),k-i+1)+i;\n t := AA[i];\n \+ AA[i] := AA[rdm];\n AA[rdm] := t\n end do;\n AA\n end proc;\n\n objects := [seq(i,i=1..numobjects)];\n DD := Arra y(1..numobjects);\n for j to numtrials do\n shuff := randarrang e(objects);\n c := 0;\n for i to numobjects do\n if s huff[i]=objects[i] then c:=c+1 end if;\n end do;\n if c>0 th en DD[c]:=DD[c]+1 end if; \n end do:\n convert(DD,list);\nend pr oc:\n\nL := proc(k::posint)\n local i;\n option remember;\n if k =1 then 0 elif k=2 then 1 else\n (k-1)*(L(k-1)+L(k-2));\n end i f;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "Here are the results of some simulations for determining the experimental probability that a random permutation is complete, t hat is, it does not leave any object in its original position.." }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 145 ": After the experime ntal probability is given, the theoretical probabilty is calculated fo r comparison. This requires that one of the procedures " }{TEXT 0 1 "L " }{TEXT -1 38 " from the previous sections is active." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "trial 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "100000 trials with 3 letters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "randomize():\nnumletters := 3;\nnumtrials := 100000;\ndistrib : = trial_perms(numletters,numtrials);\nnon_static_trials := numtrials-a dd(distrib[i],i=1..numletters);\nprob_non_static := non_static_trials/ numtrials;\nevalf(%);\nevalf(L(numletters)/numletters!);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%+numlettersG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*numtrialsG\"'++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%(distribG7%\"&e,&\"\"!\"&7l\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2 non_static_trialsG\"&IL$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0prob_no n_staticG#\"%LL\"&++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++LL!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLLLL!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "trial 2" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "100000 trials with 4 letters." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "randomize():\nnumletters := 4;\nnumtrials := 100000;\ndistrib := \+ trial_perms(numletters,numtrials);\nnon_static_trials := numtrials-add (distrib[i],i=1..numletters);\nprob_non_static := non_static_trials/nu mtrials;\nevalf(%);\nevalf(L(numletters)/numletters!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+numlettersG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*numtrialsG\"'++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(distri bG7&\"&PL$\"&(*[#\"\"!\"%gT" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2non_ static_trialsG\"&1w$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0prob_non_st aticG#\"&.)=\"&++&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++ggP!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++]P!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "trial 3" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 23 " trials with 5 letters." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "randomize():\nnumletters := 5;\nnumtrials := 10^6;\ndistrib := tr ial_perms(numletters,numtrials);\nnon_static_trials := numtrials-add(d istrib[i],i=1..numletters);\nprob_non_static := non_static_trials/numt rials;\nevalf(%);\nevalf(L(numletters)/numletters!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+numlettersG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*numtrialsG\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(distrib G7'\"'>ZP\"'\\p;\"&*f$)\"\"!\"%@%)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%2non_static_trialsG\"'7jO" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0prob _non_staticG#\"&*yX\"'+]7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++7jO !#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nmmmO!#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "trial 4" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 23 " trials with 6 lett ers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "randomize():\nnumletters := 6;\nnumtrials := 10^6;\n distrib := trial_perms(numletters,numtrials);\nnon_static_trials := nu mtrials-add(distrib[i],i=1..numletters);\nprob_non_static := non_stati c_trials/numtrials;\nevalf(%);\nevalf(L(numletters)/numletters!);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+numlettersG\"\"'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*numtrialsG\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(distribG7(\"'OrO\"'lp=\"&pb&\"&f2#\"\"!\"%<9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2non_static_trialsG\"'a\"o$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0prob_non_staticG#\"'xS=\"'++]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++a\"o$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+c bb!o$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "trial 5" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 23 " tr ials with 7 letters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "randomize():\nnumletters := 7;\nnumtrial s := 10^6;\ndistrib := trial_perms(numletters,numtrials);\nnon_static_ trials := numtrials-add(distrib[i],i=1..numletters);\nprob_non_static \+ := non_static_trials/numtrials;\nevalf(%);\nevalf(L(numletters)/numlet ters!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+numlettersG\"\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*numtrialsG\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(distribG7)\"'SyO\"')G$=\"&)*>'\"&pP\"\"%OT\"\"! \"$1#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2non_static_trialsG\"'j(o$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0prob_non_staticG#\"'j(o$\"(+++ \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++j(o$!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+H9dyO!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "trial 6" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "5*`.`*10^6;" "6#*(\"\"&\"\"\"%\".GF%\"#5\"\"'" } {TEXT -1 23 " trials with 7 letters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "randomize():\nnumletters \+ := 7;\nnumtrials := 5*10^6;\ndistrib := trial_perms(numletters,numtria ls);\nnon_static_trials := numtrials-add(distrib[i],i=1..numletters); \nprob_non_static := non_static_trials/numtrials;\nevalf(%);\nevalf(L( numletters)/numletters!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+numlet tersG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*numtrialsG\"(+++&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(distribG7)\"('QR=\"')f<*\"'&*GJ\"&@ 'p\"&p3#\"\"!\"%P5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2non_static_tr ialsG\"(%fQ=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0prob_non_staticG#\" '(H>*\"(++]#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++!)=xO!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+H9dyO!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 34 "The inclusion-exclusion principle " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 15 "Given two s ets " }{TEXT 293 1 "A" }{TEXT -1 5 " and " }{TEXT 294 1 "B" }{TEXT -1 6 ", let " }{XPPEDIT 18 0 "`union`(A,B);" "6#-%&unionG6$%\"AG%\"BG" } {TEXT -1 14 " denote their " }{TEXT 259 5 "union" }{TEXT -1 10 ", and \+ let " }{XPPEDIT 18 0 "`intersect`(A,B);" "6#-%*intersectG6$%\"AG%\"BG " }{TEXT -1 14 " denote their " }{TEXT 259 12 "intersection" }{TEXT -1 60 ". Let the number of elements of a finite set be denoted by " } {XPPEDIT 18 0 "abs(A)" "6#-%$absG6#%\"AG" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 20 "For two finite sets " }{TEXT 295 1 "A" }{TEXT -1 5 " and " }{TEXT 296 1 "B" }{TEXT -1 9 " we have " }}{PARA 259 "" 0 " " {TEXT -1 3 " | " }{XPPEDIT 18 0 "`union`(A,B);" "6#-%&unionG6$%\"AG% \"BG" }{TEXT -1 2 " |" }{XPPEDIT 18 0 "`` = abs(A)+abs(B)-abs(`interse ct`(A,B));" "6#/%!G,(-%$absG6#%\"AG\"\"\"-F'6#%\"BGF*-F'6#-%*intersect G6$F)F-!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "For thre e finite sets " }{TEXT 297 1 "A" }{TEXT -1 2 ", " }{TEXT 298 1 "B" } {TEXT -1 5 " and " }{TEXT 299 1 "C" }{TEXT -1 1 " " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = abs(`union`(A,B))+abs(C)-abs(`in tersect`(`union`(A,B),C));" "6#/%!G,(-%$absG6#-%&unionG6$%\"AG%\"BG\" \"\"-F'6#%\"CGF.-F'6#-%*intersectG6$-%&unionG6$%\"AG%\"BG%\"CG!\"\"" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "`` = abs(A)+abs(B)-abs(` intersect`(A,B))+abs(C)-abs(`intersect`(`union`(A,B),C));" "6#/%!G,,-% $absG6#%\"AG\"\"\"-F'6#%\"BGF*-F'6#-%*intersectG6$%\"AG%\"BG!\"\"-F'6# %\"CGF*-F'6#-%*intersectG6$-%&unionG6$%\"AG%\"BG%\"CGF5" }{TEXT -1 17 " ------- (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " Since " }{XPPEDIT 18 0 "`intersect`(`union`(A,B),C) = `un ion`(`intersect`(A,C),`intersect`(B,C));" "6#/-%*intersectG6$-%&unionG 6$%\"AG%\"BG%\"CG-%&unionG6$-%*intersectG6$%\"AG%\"CG-F16$%\"BGF4" } {TEXT -1 10 ", we have " }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(`intersect`(`union`(A,B),C)) = abs(`union`(`intersect`(A,C), `intersect`(B,C)) )" "6#/-%$absG6#-%*intersectG6$-%&unionG6$%\"AG%\"BG %\"CG-F%6#-F+6$-F(6$F-F/-F(6$F.F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ abs(`intersect`(A,C))+abs(`intersect`(B,C))-``;" "6#/%!G,(-%$absG6#-%* intersectG6$%\"AG%\"CG\"\"\"-F'6#-F*6$%\"BGF-F.F$!\"\"" }{TEXT 304 1 " |" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`intersect`(A,B)" "6#-%*intersectG6 $%\"AG%\"BG" }{XPPEDIT 18 0 "`intersect`(``,C)" "6#-%*intersectG6$%!G% \"CG" }{TEXT -1 1 " " }{TEXT 305 1 "|" }{TEXT -1 15 " ------- (iv)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "It foll ows from (iii) and (iv) that " }}{PARA 263 "" 0 "" {TEXT -1 1 " " } {TEXT 300 1 "|" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`union`(A,B);" "6#-%&u nionG6$%\"AG%\"BG" }{XPPEDIT 18 0 "`union`(``,C);" "6#-%&unionG6$%!G% \"CG" }{TEXT -1 1 " " }{TEXT 301 1 "|" }{XPPEDIT 18 0 " ``= abs(A)+ab s(B)+abs(C)-abs(A intersect B)-abs(A intersect C)-abs(B intersect C)+` `" "6#/%!G,0-%$absG6#%\"AG\"\"\"-F'6#%\"BGF*-F'6#%\"CGF*-F'6#-%*inters ectG6$F)F-!\"\"-F'6#-F46$F)F0F6-F'6#-F46$F-F0F6F$F*" }{TEXT 302 1 "|" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`intersect`(A,B)" "6#-%*intersectG6$% \"AG%\"BG" }{XPPEDIT 18 0 "`intersect`(``,C)" "6#-%*intersectG6$%!G%\" CG" }{TEXT -1 1 " " }{TEXT 303 1 "|" }{TEXT -1 2 ". 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$$!+I-[#e\"FhimF`u7$$!+OMr3;FhimFeu7$$!+j`fF;FhimFju7$$!+3B)*Q;FhimFhd n7$$!+5wyU;FhimFedn7$F\\goF`dn7$FifoF[dn7$FffoFfcn7$FcfoFacn7$F`foF\\c n7$F]foFgbn7$FjeoFbbn7$FgeoF]bn7$FdeoFhan7$FaeoF[_n7$F^eoF^_n7$F[eoFa_ n7$FhdoFd_n7$FjoFg_n7$F_pFj_n7$FdpF]`n7$FipF``n7$F^qFc`n7$FcqF``n7$Ffq F]`n7$FiqFj_n7$F\\rFg_n7$F_rFd_n7$FdrFa_n7$FirF^_nFefnFj^n7$FesFhan7$F jsF]bn7$F_tFbbn7$FdtFgbn7$FitF\\cn7$F^uFacn7$FcuFfcn7$FhuF[dn7$$!+%)32 MNF0F`dn7$$!+.R7sNF0Fedn7$FcioFhdnFguF-F3F8F=FBFGFLFQFVFenFjnF_oFdo-F] v6&F_vF`^nFfhmF^^nFev-F$6%7F7$Fh\\mFh^nFj[lFfenFienF\\fnF]fnF^fnF_fnF` fnFafnFbfnFcfnFdfnFefnFj^nFjhoF[ioF\\ioF]ioF^ioF_ioF`ioFaioFbioFeioFhi oFguF-Fj\\mF_]mFd]mFj]mF]^mF`^mFc^mFg[l-F]v6&F_vFh]lFf]lFh]lFev-%%TEXT G6$7$Fh\\mFh]lQ\"A6\"-Fbjo6$7$Fd]l$!\"&FdvQ\"BFfjo-Fbjo6$7$$FhimFdvFjj oQ\"CFfjo-%+AXESLABELSG6%Q!FfjoFe[p-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6# %%NONEG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$Fi[pFi[p" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1 0" "Curve 11" "Curve 12" "Curve 13" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Fo r four finite sets " }{TEXT 316 1 "A" }{TEXT -1 2 ", " }{TEXT 317 1 "B " }{TEXT -1 2 ", " }{TEXT 318 1 "C" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "` D`" "6#%#~DG" }{TEXT -1 3 ", " }}{PARA 265 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 306 1 "|" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`union`(A,B);" "6#- %&unionG6$%\"AG%\"BG" }{XPPEDIT 18 0 "`union`(``,C);" "6#-%&unionG6$%! G%\"CG" }{XPPEDIT 18 0 "`union`(``,`D `);" "6#-%&unionG6$%!G%#D~G" } {TEXT -1 1 " " }{TEXT 307 1 "|" }{XPPEDIT 18 0 "`` = abs(A)+abs(B)+abs (C)+abs(`D `)-abs(`intersect`(A,B))-abs(`intersect`(A,C))-abs(`interse ct`(A,`D `))-abs(`intersect`(B,C))-abs(`intersect`(B,`D `))-abs(`inter sect`(C,`D `));" "6#/%!G,6-%$absG6#%\"AG\"\"\"-F'6#%\"BGF*-F'6#%\"CGF* -F'6#%#D~GF*-F'6#-%*intersectG6$F)F-!\"\"-F'6#-F76$F)F0F9-F'6#-F76$F)% #D~GF9-F'6#-F76$F-F0F9-F'6#-F76$F-%#D~GF9-F'6#-F76$F0%#D~GF9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 3 " + " }{TEXT 308 1 "|" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`intersect`(A ,B)" "6#-%*intersectG6$%\"AG%\"BG" }{XPPEDIT 18 0 "`intersect`(``,C)" "6#-%*intersectG6$%!G%\"CG" }{TEXT -1 1 " " }{TEXT 309 5 "| + |" } {TEXT -1 1 " " }{XPPEDIT 18 0 "`intersect`(A,B)" "6#-%*intersectG6$%\" AG%\"BG" }{XPPEDIT 18 0 "`intersect`(``,`D `);" "6#-%*intersectG6$%!G% #D~G" }{TEXT -1 1 " " }{TEXT 311 5 "| + |" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`intersect`(A,C);" "6#-%*intersectG6$%\"AG%\"CG" }{XPPEDIT 18 0 "`intersect`(``,`D `);" "6#-%*intersectG6$%!G%#D~G" }{TEXT -1 1 " " } {TEXT 313 5 "| + |" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`intersect`(B,C); " "6#-%*intersectG6$%\"BG%\"CG" }{XPPEDIT 18 0 "`intersect`(``,`D `); " "6#-%*intersectG6$%!G%#D~G" }{TEXT -1 1 " " }{TEXT 315 2 "| " } {TEXT -1 1 "-" }{TEXT 322 2 " |" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`inte rsect`(A,B);" "6#-%*intersectG6$%\"AG%\"BG" }{XPPEDIT 18 0 "`intersect `(``,C);" "6#-%*intersectG6$%!G%\"CG" }{XPPEDIT 18 0 "`intersect`(``,` D `);" "6#-%*intersectG6$%!G%#D~G" }{TEXT -1 1 " " }{TEXT 320 1 "|" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{TEXT 332 1 "n" }{TEXT -1 13 " finite sets " } {XPPEDIT 18 0 "A[1],A[2],` . . . `,A[n]" "6&&%\"AG6#\"\"\"&F$6#\"\"#%( ~.~.~.~G&F$6#%\"nG" }{TEXT -1 45 ", we have the following general form for the " }{TEXT 259 29 "inclusion-exclusion principle" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 321 1 "|" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`union`(A[1],A[2]);" "6#-%&unionG6$&%\"AG6#\"\"\"&% \"AG6#\"\"#" }{XPPEDIT 18 0 "` . . . `;" "6#%(~.~.~.~G" }{XPPEDIT 18 0 "`union`(``,A[n]);" "6#-%&unionG6$%!G&%\"AG6#%\"nG" }{TEXT -1 1 " " }{TEXT 323 1 "|" }{XPPEDIT 18 0 " ``= Sum(abs(A[i]),i) " "6#/%!G-%$Su mG6$-%$absG6#&%\"AG6#%\"iGF." }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"F$ !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(abs(A[i] intersect A[j]),`i " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "Using the \+ inclusion-exclusion principle to obtain a formula for " }{XPPEDIT 18 0 "L(k)" "6#-%\"LG6#%\"kG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "A[1]" "6#&%\"AG6#\"\"\"" }{TEXT -1 43 " be the set of permutations \+ of the numbers " }{XPPEDIT 18 0 "1,2,3,4,5;" "6'\"\"\"\"\"#\"\"$\"\"% \"\"&" }{TEXT -1 55 " in which the number 1 remains in the1st position . Let " }{XPPEDIT 18 0 "A[2]" "6#&%\"AG6#\"\"#" }{TEXT -1 31 " be the \+ set of permutations of " }{XPPEDIT 18 0 "1,2,3,4,5;" "6'\"\"\"\"\"#\" \"$\"\"%\"\"&" }{TEXT -1 59 " in which the number 2 remains in the 2nd position. Define " }{XPPEDIT 18 0 "A[3], A[4]" "6$&%\"AG6#\"\"$&F$6# \"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "A[5]" "6#&%\"AG6#\"\"&" } {TEXT -1 18 " in a similar way." }}{PARA 0 "" 0 "" {TEXT -1 89 "The se t of permutations in which at least one number remains in its original position is " }{XPPEDIT 18 0 "`union`(A[1],A[2])" "6#-%&unionG6$&%\"A G6#\"\"\"&F'6#\"\"#" }{XPPEDIT 18 0 " `union`(``,A[3])" "6#-%&unionG6 $%!G&%\"AG6#\"\"$" }{XPPEDIT 18 0 "`union`(``,A[4])" "6#-%&unionG6$%!G &%\"AG6#\"\"%" }{XPPEDIT 18 0 "`union`(``,A[5])" "6#-%&unionG6$%!G&%\" AG6#\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 78 "The number \+ of permutations for which no number is in its original position is " } {XPPEDIT 18 0 "L(5) = 5!-``;" "6#/-%\"LG6#\"\"&,&-%*factorialG6#F'\"\" \"%!G!\"\"" }{TEXT -1 2 "| " }{XPPEDIT 18 0 "`union`(A[1],A[2])" "6#-% &unionG6$&%\"AG6#\"\"\"&F'6#\"\"#" }{XPPEDIT 18 0 " `union`(``,A[3]) " "6#-%&unionG6$%!G&%\"AG6#\"\"$" }{XPPEDIT 18 0 "`union`(``,A[4])" "6 #-%&unionG6$%!G&%\"AG6#\"\"%" }{XPPEDIT 18 0 "`union`(``,A[5])" "6#-%& unionG6$%!G&%\"AG6#\"\"&" }{TEXT -1 5 " |. " }}{PARA 0 "" 0 "" {TEXT -1 37 "By the inclusion-exclusion principle " }}{PARA 0 "" 0 "" {TEXT -1 3 " | " }{XPPEDIT 18 0 "`union`(A[1],A[2])" "6#-%&unionG6$&%\"AG6# \"\"\"&F'6#\"\"#" }{XPPEDIT 18 0 " `union`(``,A[3])" "6#-%&unionG6$%! G&%\"AG6#\"\"$" }{XPPEDIT 18 0 "`union`(``,A[4])" "6#-%&unionG6$%!G&% \"AG6#\"\"%" }{XPPEDIT 18 0 "`union`(``,A[5])" "6#-%&unionG6$%!G&%\"AG 6#\"\"&" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "`` = Sum(abs(A[i]),i);" "6# /%!G-%$SumG6$-%$absG6#&%\"AG6#%\"iGF." }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(abs (A[i]*`^`*A[j]),`i " 0 "" {MPLTEXT 1 0 246 "e := evalf(exp(1),30):\nprintf(\"\\n k \+ k! L(k) p(k)\\n\\n\");\nfor k from 1 to 20 do\n t := k!;\n s := round(evalf(k!/e,30));\n p := ev alf(s/t,20); \n printf(\"%4d %23d%23d%25.16f\\n\",k,t,s,p);\nend do: " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 69 " k \+ k! L(k) p(k)" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 76 " 1 \+ 1 0 0.0000000000000000" } }{PARA 6 "" 1 "" {TEXT -1 76 " 2 2 \+ 1 .5000000000000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " \+ 3 6 2 .33333333333 33333" }}{PARA 6 "" 1 "" {TEXT -1 76 " 4 24 \+ 9 .3750000000000000" }}{PARA 6 "" 1 "" {TEXT -1 76 " 5 120 44 .366 6666666666667" }}{PARA 6 "" 1 "" {TEXT -1 76 " 6 \+ 720 265 .3680555555555556" }}{PARA 6 "" 1 " " {TEXT -1 76 " 7 5040 1854 \+ .3678571428571429" }}{PARA 6 "" 1 "" {TEXT -1 76 " 8 \+ 40320 14833 .3678819444444444" }}{PARA 6 "" 1 "" {TEXT -1 76 " 9 362880 13 3496 .3678791887125220" }}{PARA 6 "" 1 "" {TEXT -1 76 " 10 \+ 3628800 1334961 .3678794642857143" }}{PARA 6 "" 1 "" {TEXT -1 76 " 11 39916800 \+ 14684570 .3678794392336059" }}{PARA 6 "" 1 "" {TEXT -1 76 " 12 479001600 176214841 .3678794413 212816" }}{PARA 6 "" 1 "" {TEXT -1 76 " 13 6227020800 \+ 2290792932 .3678794411606912" }}{PARA 6 "" 1 "" {TEXT -1 76 " 14 87178291200 32071101049 \+ .3678794411721619" }}{PARA 6 "" 1 "" {TEXT -1 76 " 15 130 7674368000 481066515734 .3678794411713972" }}{PARA 6 "" 1 "" {TEXT -1 76 " 16 20922789888000 76970642517 45 .3678794411714450" }}{PARA 6 "" 1 "" {TEXT -1 76 " 17 \+ 355687428096000 130850092279664 .3678794411714422" }} {PARA 6 "" 1 "" {TEXT -1 76 " 18 6402373705728000 235530 1661033953 .3678794411714423" }}{PARA 6 "" 1 "" {TEXT -1 76 " \+ 19 121645100408832000 44750731559645106 .367879441171 4423" }}{PARA 6 "" 1 "" {TEXT -1 76 " 20 2432902008176640000 \+ 895014631192902121 .3678794411714423" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 81 "A procedure for constructing pictorial re presentations of complete permutations: " }{TEXT 0 8 "permdiag" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "permdiag: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 286 18 "Calling S equence:\n" }}{PARA 0 "" 0 "" {TEXT -1 69 " permdiag(A, . . options . . ) or permdiag(n, . . options . . ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 " " {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 76 " A - a list o f positive integers to represent a complete permutation" }}{PARA 0 "" 0 "" {TEXT -1 107 " n - a positive integer to specify the list [1, 2, . . . , n] for which to construct an animation" }}{PARA 0 " " 0 "" {TEXT -1 76 " which illustrates all the complete \+ permutations of this list " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }} {PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "per mdiag" }{TEXT -1 246 " constructs a single diagram based on a regular \+ polygon to provide a pictorial representation of a complete permutatio n or a sequence of diagrams which form an animation to illustrate all \+ the complete permutations of the list [1, 2, . . . , n].\n" }}{PARA 0 "" 0 "" {TEXT 287 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 14 "colour/color=c" }}{PARA 0 "" 0 "" {TEXT -1 94 "This optio n allows the background colour of the polygon used in the illistration to be chosen." }}{PARA 0 "" 0 "" {TEXT -1 99 "In addition to the usua l Maple plotting colour options, suitable light colours can be chosen \+ from: " }}{PARA 0 "" 0 "" {TEXT -1 69 " light_blue, light_green, ligh t_orange, light_magenta, light_purple." }}{PARA 0 "" 0 "" {TEXT -1 35 "The default colour is light_purple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "insequence=true/false" }}{PARA 0 "" 0 " " {TEXT -1 291 "In the case where the first argument is a positive int eger, this option can be used to determine whether the diagrams which \+ illustrate all the complete permutations on [1, 2, . . . , n] are disp layed so as to from an animation or are all displayed at once. The def ault is \"insequence=true\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "delta=r " }}{PARA 0 "" 0 "" {TEXT -1 190 " When drawing \"double arrows\" to indicate transpositions the spacing \+ of the two arrows can be controlled with this option.\n\"delta\" must \+ evaluate to a floating point number between 0 and 0.1. " }}{PARA 0 "" 0 "" {TEXT -1 28 "The default is \"delta=0.03\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "in_radius=r\nThe radius \+ of the circle used for the numbering of the vertices of the regular po lygon is controlled with this option." }}{PARA 0 "" 0 "" {TEXT -1 97 " It gives the radius as a ratio of the radius of the circumscribing cir cle of the regular polygon." }}{PARA 0 "" 0 "" {TEXT -1 75 "\"in_radiu s\" must evaluate to a floating point number between 0.7 and 0.95. " } }{PARA 0 "" 0 "" {TEXT -1 32 "The default is \"in_radius=0.85\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "out_radi us=r\nThe radius of the circle used for the numbering of the vertices \+ of the regular polygon is controlled with this option." }}{PARA 0 "" 0 "" {TEXT -1 97 "It gives the radius as a ratio of the radius of the \+ circumscribing circle of the regular polygon." }}{PARA 0 "" 0 "" {TEXT -1 74 "\"out_radius\" must evaluate to a floating point number b etween 1.1 and 1.5." }}{PARA 0 "" 0 "" {TEXT -1 34 "The default is \"o ut_radius=1.13\".\n" }}{PARA 0 "" 0 "" {TEXT -1 11 "thickness=n" }} {PARA 0 "" 0 "" {TEXT -1 52 "The thickness of the arrow - default: \"t hickness=2\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "head_height=hh" }}{PARA 0 "" 0 "" {TEXT -1 191 "The heigh t of the head of the arrow as a ratio of the radius of the circumscrib ing circle of the regular polygon.\n\"head_height\" must evaluate to a floating point number between 0.05 and 0.2. " }}{PARA 0 "" 0 "" {TEXT -1 34 "The default is \"head_height=0.1\".\n" }}{PARA 0 "" 0 "" {TEXT -1 13 "head_width=hw" }}{PARA 0 "" 0 "" {TEXT -1 112 "The width \+ of the head of the arrow.\n\"head_width\" must evaluate to a floating \+ point number between 0.05 and 0.2. " }}{PARA 0 "" 0 "" {TEXT -1 32 "Th e default is \"head_width=0.1\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to a ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure activ e open the subsection, place the cursor anywhere after the prompt [ > \+ and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "permdiag: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(comb struct):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5551 "permdiag := proc(ff)\n local n,del,A,B,allp,ct,di ags,p,complete,\n startopts,Options,i,j,k,clr,r1,r2,inseq,thk,hh,hw; \n\n if nargs>1 and type(args[2],posint) then\n startopts := 3; \n n := args[2];\n else\n startopts := 2;\n n := 0;\n end if;\n\n Digits := 10;\n r1 := 0.85;\n r2 := 1.1;\n del \+ := 0.03;\n thk := 2;\n hh := 0.1;\n hw := 0.1;\n clr := COLOR( RGB,.9,.7,1);\n inseq := true;\n if nargs>=startopts then\n O ptions:=[args[startopts..nargs]];\n if not type(Options,list(equa tion)) then\n error \"each optional argument must be an equati on\"\n end if;\n if hasoption(Options,insequence,'inseq','Op tions') then\n if inseq<>true then inseq := false end if;\n \+ end if;\n if hasoption(Options,thickness,'thk','Options') then \n if not type(thk,posint) then\n error \"\\\"thick ness\\\" must be a positive integer\"\n end if;\n end if ;\n if hasoption(Options,delta,'del','Options') then\n de l := evalf(del);\n if not type(del,float) or del<0 or del>0.1 \+ then\n error \"\\\"delta\\\" must evaluate to a positive fl oat between 0 and 0.1\"\n end if;\n end if;\n if has option(Options,in_radius,'r1','Options') then\n r1 := evalf(r1 );\n if not type(r1,float) or r1<0.7 or r1>0.95 then\n \+ error \"\\\"out_radius\\\" must evaluate to a positive float betwe en 0.7 and 0.95\"\n end if;\n end if;\n if hasoption (Options,out_radius,'r2','Options') then\n r2 := evalf(r2);\n \+ if not type(r2,float) or r2<1.05 or r2>1.5 then\n e rror \"\\\"out_radius\\\" must evaluate to a positive float between 1. 05 and 1.5\"\n end if;\n end if;\n if hasoption(Opti ons,head_height,'hh','Options') then\n hh := evalf(hh);\n \+ if not type(hh,float) or hh<0.05 or hh>0.2 then\n error \"\\\"head_height\\\" must evaluate to a positive float between 0.05 \+ and 0.2\"\n end if;\n end if;\n if hasoption(Options ,head_width,'hw','Options') then\n hw := evalf(hw);\n \+ if not type(hw,float) or hw<0.05 or hw>0.2 then\n error \" \\\"head_width\\\" must evaluate to a positive float between 0.05 and \+ 0.2\"\n end if;\n end if;\n if hasoption(Options,col or,'clr','Options') or\n hasoption(Options,colour,'clr','Optio ns') then\n if clr=light_magenta then\n clr := COLO R(RGB,1,.75,1)\n elif clr=light_purple then\n clr : = COLOR(RGB,.9,.7,1)\n elif clr=light_green then\n \+ clr := COLOR(RGB,.8,1,.8) \n elif clr=light_blue then\n \+ clr := COLOR(RGB,.8,.8,1)\n elif clr=light_orange then\n clr := COLOR(RGB,1,.9,.65)\n end if; \n en d if;\n if nops(Options)>0 then\n error \"%1 is not a val id option for %2\",op(1,Options),procname;\n end if;\n end if; \n \n if type(ff,posint) then k := ff;\n A := [seq(i,i=1..k)] ;\n print(A);\n allp := traperror(iterstructs(Permutation(A) ));\n if has(allp,iterstructs) then\n error \"please load the package: \\\"combstruct\\\"\"\n end if;\n ct := 0;\n \+ diags := [];\n while not finished(allp) do\n p := next struct(allp);\n complete := true;\n for i to k do\n \+ if p[i]=A[i] then\n complete := false;\n \+ break;\n end if;\n end do; \n if \+ complete then\n ct := ct+1;\n diags := [op(diags ),\n permdiag_draw(p,ct,del,r1,r2,clr,thk,hh,hw)];\n \+ end if; \n end do;\n return plots[display](diags,inse quence=inseq,\n scaling=constrained);\n elif type(ff,'lis t(posint)') then\n A := ff;\n k := nops(A);\n B := []; \n for i to k do\n j := A[i];\n if i=j then\n \+ error \"%1 does not represent a complete permutation\",A;\n \+ end if;\n if member(j,B) then\n error \"%1 mu st not contain repeated numbers\",A;\n end if;\n B := \+ [op(B),j];\n end do;\n permdiag_draw(A,n,del,r1,r2,clr,thk,h h,hw);\n end if;\nend proc: # of permdiag\n\npermdiag_draw := proc(A ,n,del,r1,r2,clr,thk,hh,hw)\n local k,B,c,d,V,nums,txt,pts,p1,p2,i,j ,u,\n arws,sn,cs,x,y,h,delx,dely,tt,arlen;\n k := nops(A);\n d : = evalf(2*Pi/k);\n c := evalf(1/k);\n V := [seq([cos(d*i),sin(d*i) ],i=1..k)];\n p1 := plots[polygonplot](V,color=clr);\n p2 := NULL; \n arws := [];\n nums := [];\n for i to k do\n j := A[i];\n arws := [op(arws),[i,j]];\n if member([j,i],arws) then\n \+ x := V[j,1]+V[i,1];\n y := V[j,2]+V[i,2];\n h := sqrt(x^2+y^2);\n if h>1e-8 then \n sn := y/h;\n \+ cs := x/h;\n else\n sn := (V[j,1]-V[i,1])/ 2;\n cs := (V[i,2]-V[j,2])/2;\n end if; \n \+ delx := del*cs;\n dely := del*sn;\n pts := [[r1*(V[i, 1]-delx),r1*(V[i,2]-dely)],\n [r1*(V[j,1]-delx),r1*(V[j ,2]-dely)]];\n else\n pts := map(u->r1*u,[V[i],V[j]]);\n \+ end if;\n arlen := sqrt((V[j,1]-V[i,1])^2+(V[i,2]-V[j,2])^2) ;\n p2 := p2,plottools[arrow](op(pts),0,hw,hh/arlen,arrow,\n \+ color=COLOR(HUE,i*c),thickness=thk);\n nums := [op(nums),[r2 *V[i,1],r2*V[i,2],i]];\n end do;\n txt := plots[textplot](nums);\n if n<>0 then\n tt := cat(convert(n,string),\" \",convert(A,s tring),\" \");\n else\n tt := convert(A,string)\n end if; \+ \n plots[display]([p1,p2,txt],scaling=constrained,\n \+ axes=none,title=tt);\nend proc: # of permdiag_draw" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examp les are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT 0 8 "permdiag" }{TEXT -1 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 41 "Examples illustrating single permutations" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 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}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 256 " " 0 "" {TEXT 266 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 20 "code for \+ 1st picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 493 "p1 := plots[polygonplot]([[0,0],[1,0],[1,2],[0,2]] ,color=COLOR(RGB,.95,.85,1)):\np2 := plots[polygonplot]([[2,0],[3,0],[ 3,2],[2,2]],color=COLOR(RGB,1,.85,.95)):\np3 := plots[polygonplot]([[4 ,0],[5,0],[5,2],[4,2]],color=COLOR(RGB,.95,1,.85)):\nt1 := plots[textp lot]([.5,1,`A`],color=COLOR(RGB,.65,0,.8)):\nt2 := plots[textplot]([2. 5,1,`B`],color=COLOR(RGB,.8,0,.65)):\nt3 := plots[textplot]([4.5,1,`C` ],color=COLOR(RGB,.65,.8,0)):\nplots[display]([p1,p2,p3,t1,t2,t3],axes =none,font=[TIMES,ROMAN,48]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 20 "code for 2nd picture" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 641 "p1 := plots[polygonplot ]([[0,0],[1,0],[1,2],[0,2]],color=COLOR(RGB,.95,.85,1)):\np2 := plots[ polygonplot]([[2,0],[3,0],[3,2],[2,2]],color=COLOR(RGB,1,.85,.95)):\np 3 := plots[polygonplot]([[4,0],[5,0],[5,2],[4,2]],color=COLOR(RGB,.95, 1,.85)):\np4 := plots[polygonplot]([[6,0],[7,0],[7,2],[6,2]],color=COL OR(RGB,1,.95,.85)):\nt1 := plots[textplot]([.5,1,`A`],color=COLOR(RGB, .65,0,.8)):\nt2 := plots[textplot]([2.5,1,`B`],color=COLOR(RGB,.8,0,.6 5)):\nt3 := plots[textplot]([4.5,1,`C`],color=COLOR(RGB,.65,.8,0)):\nt 4 := plots[textplot]([6.5,1,`D`],color=COLOR(RGB,.8,.65,0)):\nplots[di splay]([p1,p2,p3,p4,t1,t2,t3,t4],axes=none,font=[TIMES,ROMAN,48]);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 22 "code for Venn diagr am " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1819 "with(plots):\nd := Pi/36:\nr := evalf(cos(5*Pi/18)* 2/sqrt(3)):\na := evalf(r*sin(Pi/3)): b := evalf(r*cos(Pi/3)):\nc1 := \+ evalf([seq([cos(d*i),sin(d*i)+r],i=0..72)]):\nc2 := evalf([seq([cos(d* i)+a,sin(d*i)-b],i=0..72)]):\nc3 := evalf([seq([cos(d*i)-a,sin(d*i)-b] ,i=0..72)]):\ns1 := evalf(seq([cos(d*i),sin(d*i)+r],i=-2..38)):\ns2 := evalf(seq([cos(d*(22-i))-a,sin(d*(22-i))-b],i=0..12)):\ns3 := evalf(s eq([cos(d*(26-i))+a,sin(d*(26-i))-b],i=0..12)):\ns4 := evalf(seq([cos( d*i)+a,sin(d*i)-b],i=-26..14)):\ns5 := evalf(seq([cos(d*(-i)),sin(d*(- i))+r],i=2..14)):\ns6 := evalf(seq([cos(d*(-i))-a,sin(d*(-i))-b],i=-2. .10)):\ns7 := evalf(seq([cos(d*i)-a,sin(d*i)-b],i=22..62)):\ns8 := eva lf(seq([cos(d*(46-i))+a,sin(d*(46-i))-b],i=0..12)):\ns9 := evalf(seq([ cos(d*(50-i)),sin(d*(50-i))+r],i=0..12)):\ns10 := evalf(seq([cos(d*i), sin(d*i)+r],i=50..58)):\ns11 := evalf(seq([cos(d*i)-a,sin(d*i)-b],i=2. .10)):\ns12 := evalf(seq([cos(d*i)+a,sin(d*i)-b],i=26..34)):\nm1 := [[ 0,1],s1,s2,s3]:\nm2 := [[.96,-.6],s4,s5,s6]:\nm3 := [[-.96,-.6],s7,s8, s9]:\nm4 := [[.56,.32],s3,s5,s11]:\nm5 := [[-.56,.32],s9,s2,s12]:\nm6 \+ := [[0,-.6],s6,s8,s10]:\nm7 := [[0,0],s10,s11,s12]:\np1 := plot(c1,col or=grey):\np2 := plot(c2,color=grey):\np3 := plot(c3,color=grey):\np4 \+ := polygonplot(m1,color=COLOR(RGB,1,.85,.95),style=PATCHNOGRID):\np5 : = polygonplot(m2,color=COLOR(RGB,.95,.85,1),style=PATCHNOGRID):\np6 := polygonplot(m3,color=COLOR(RGB,.95,1,.85),style=PATCHNOGRID):\np7 := \+ polygonplot(m4,color=COLOR(RGB,.9,.7,.8),style=PATCHNOGRID):\np8 := po lygonplot(m5,color=COLOR(RGB,.75,.75,.6),style=PATCHNOGRID):\np9 := po lygonplot(m6,color=COLOR(RGB,.8,.7,.8),style=PATCHNOGRID):\np10 := pol ygonplot(m7,color=COLOR(RGB,.75,.55,.65),style=PATCHNOGRID):\nt1 := te xtplot([[0,.8,`A`],[.9,-.5,`B`],[-.9,-.5,`C`]]):\ndisplay(\{p1,p2,p3,p 4,p5,p6,p7,p8,p9,p10,t1\},scaling=constrained,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }