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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 27 "TD : Entiers - d\351nom
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"nG\"\"\",&F%F&F&F&F&,&F%\"\"#F
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cices 2 et 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(3^(2*n+1
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "seq(2^(2*n)+15*n-1 mod 9,n=0..30);
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10 "Exercice 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sum(sum(i*
j,j=i..n),i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*&%\"nG\"\"#,&
F%\"\"\"F(F(F&#F(\"\"%*&F%F&F'F(#!\"\"F**&F%F(F'F&F)*&F%F(F'F(F,*$F'\"
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F&#!\"\"\"\"'F%#F&\"\"$#F&\"\"#F&*&F'F&F%F,F+*&F%F.F'F.F-*$F'F.#!#B\"#
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(\");" }}{PARA 11 "" 
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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "irem(1789,10),iquo(1789,
10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"*\"$y\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 58 "phi:=n->if n<=9 then n else irem(n,10)+ph
i(iquo(n,10)) fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG:6#%\"nG6
\"6$%)operatorG%&arrowGF(@%19$\"\"*F.,&-%%iremG6$F.\"#5\"\"\"-F$6#-%%i
quoGF3F5F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "phi(1515);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "N:=4444^4444:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "evalf(ln(N)/ln(10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)y
q5i\"!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 62 "La repr\351sentatio
n d\351cimale de N poss\350de donc 16211 chiffres..." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "phi(N);" }}{PARA 8 "" 1 "" {TEXT -1 44 "Er
ror, (in phi) too many levels of recursion" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 258 53 "Pas tr\350s surprenant : on va la refaire en it\351ratif
..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "phi2:=proc(n)\nloca
l reste,compt;\nreste:=n;\ncompt:=0;\nwhile reste>0 do\ncompt:=compt+i
rem(reste,10);\nreste:=iquo(reste,10) od;\nRETURN(compt+reste)\nend:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "phi2(1515);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "phi2(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&,E(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "phi2(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "phi2(
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0 "" {TEXT -1 10 "Exercice 9" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "ni:=product(365-i,i=0..44);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#
niG\"^r+++++Se./&f>%RJ(Q[R?))\\n3a')3p)\\ma.Fw*3*>'HX[X*z'**[G1XEVn0M`
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 " 0 "" {MPLTEXT 1 0 4 "1-\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&
**e(4%*!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 93 "C'est la probabili
t\351 pour qu'au moins deux personnes parmi 45 aient m\352me date anni
versaire..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "limit((n^2)!
/(n^2-2*n)!/(n^2)^(2*n),n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%$expG6#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\")
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+KGN`8!#5" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 48 "limit((n^2)!/(n^2-3*n)!/(n^2)^(3*n),n=infini
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "
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-1 33 "Permutations : exercices 11 et 14" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 36 "G\351n\351rons l'ensemble des permutations" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "insere:=(l,it,pos)->[seq(l[j],j=1..
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arning, `j` in call to `seq` is not local" }}{PARA 7 "" 1 "" {TEXT -1 
42 "Warning, `j` in call to `seq` is not local" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'insereG:6%%\"lG%#itG%$posG6\"6$%)operatorG%&arrowGF*
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "insere([12,3,5],2,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"#7\"\"#\"\"$\"\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "insere([12,3,5],2,3);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7&\"#7\"\"$\"\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "insere([12,3,5],2,4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&\"#7\"\"$\"\"&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 201 "permut:=proc(n)\nlocal prec,res;\nif n=1 then RETURN
([[1]]) fi;\nprec:=permut(n-1);\nres:=NULL;\nfor k from n to 1 by -1 d
o\n  for l from 1 to (n-1)! do res:=res,insere(prec[l],n,k) od od;\nRE
TURN([res])\nend;" }}{PARA 7 "" 1 "" {TEXT -1 41 "Warning, `k` is impl
icitly declared local" }}{PARA 7 "" 1 "" {TEXT -1 41 "Warning, `l` is \+
implicitly declared local" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'permut
G:6#%\"nG6&%%precG%$resG%\"kG%\"lG6\"F-C'@$/9$\"\"\"-%'RETURNG6#7#7#F2
>8$-F$6#,&F1F2!\"\"F2>8%%%NULLG?(8&F1F=F2%%trueG?(8'F2F2-%*factorialGF
;FC>F?6$F?-%'insereG6%&F96#FEF1FB-F46#7#F?F-F-" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "permut(4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
7:7&\"\"\"\"\"#\"\"$\"\"%7&F&F%F'F(7&F%F'F&F(7&F&F'F%F(7&F'F%F&F(7&F'F
&F%F(7&F%F&F(F'7&F&F%F(F'7&F%F'F(F&7&F&F'F(F%7&F'F%F(F&7&F'F&F(F%7&F%F
(F&F'7&F&F(F%F'7&F%F(F'F&7&F&F(F'F%7&F'F(F%F&7&F'F(F&F%7&F(F%F&F'7&F(F
&F%F'7&F(F%F'F&7&F(F&F'F%7&F(F'F%F&7&F(F'F&F%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "nops(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C
" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 37 "Nombre moyen de points fixe
s (exo 14)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "points_fixes:
=proc(l)\nlocal k,r;\nr:=0;\nfor k from 1 to nops(l) do if l[k]=k then
 r:=r+1 fi od;\nRETURN(r);\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
-points_fixesG:6#%\"lG6$%\"kG%\"rG6\"F+C%>8%\"\"!?(8$\"\"\"F2-%%nopsG6
#9$%%trueG@$/&F66#F1F1>F.,&F.F2F2F2-%'RETURNG6#F.F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "points_fixes([2,1,3,4]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 123 "total_points_fixes:=proc(n)\nlocal l,r;\nl:=permut(n);\nr:=0;
\nfor k from 1 to n! do r:=r+points_fixes(l[k]) od;\nRETURN(r)\nend;" 
}}{PARA 7 "" 1 "" {TEXT -1 41 "Warning, `k` is implicitly declared loc
al" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3total_points_fixesG:6#%\"nG6%
%\"lG%\"rG%\"kG6\"F,C&>8$-%'permutG6#9$>8%\"\"!?(8&\"\"\"F9-%*factoria
lGF2%%trueG>F5,&F5F9-%-points_fixesG6#&F/6#F8F9-%'RETURNG6#F5F,F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "total_points_fixes(5);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "seq(total_points_fixes(n),n=1..7);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6)\"\"\"\"\"#\"\"'\"#C\"$?\"\"$?(\"%S]" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "seq(k!,k=0..7);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6*\"\"\"F#\"\"#\"\"'\"#C\"$?\"\"$?(\"%S]" }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 37 "Nombre moyen de d\351rangements (exo 11)
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "nb_derangements:=proc(n
)\nlocal l,r;\nl:=permut(n);\nr:=0;\nfor k from 1 to n! do if points_f
ixes(l[k])=0 then r:=r+1 fi od;\nRETURN(r)\nend;" }}{PARA 7 "" 1 "" 
{TEXT -1 41 "Warning, `k` is implicitly declared local" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%0nb_derangementsG:6#%\"nG6%%\"lG%\"rG%\"kG6\"F,C
&>8$-%'permutG6#9$>8%\"\"!?(8&\"\"\"F9-%*factorialGF2%%trueG@$/-%-poin
ts_fixesG6#&F/6#F8F6>F5,&F5F9F9F9-%'RETURNG6#F5F,F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "nb_derangements(5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#W" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "seq(
nb_derangements(i)/i!,i=1..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)\"\"
!#\"\"\"\"\"##F%\"\"$#F(\"\")#\"#6\"#I#\"#`\"$W\"#\"$.\"\"$!G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6)\"\"!$\"+++++]!#5$\"+LLLLLF&$\"++++]PF&$\"+nmmmOF&$\"
+cbb!o$F&$\"+H9dyOF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eva
lf(exp(-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7WzyO!#5" }}}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 15" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "sum((-1)^k*binomial(n,k),k=0..n);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "sum(k*binomial(n,k),k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,$*&)\"\"#%\"nG\"\"\"F'F(#F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 39 "sum((-1)^(k+1)*k*binomial(n,k),k=0..n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sum
(k*(k-1)*binomial(n,k),k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*
()\"\"#%\"nG\"\"\"F'F(,&F'F(!\"\"F(F(#F(\"\"%" }}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 11 "Exercice 16" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "sum(k^2*binomial(n,k),k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,&*&)\"\"#%\"nG\"\"\"F'F(#F(F&*(F%F(F'F(,&F'F(!\"\"F(F(#F(\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(\");" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*(%\"nG\"\"\")\"\"#F%F&,&F%F&F&F&F&#F&\"\"%" }}}
}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 17" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 28 "sum(binomial(p+i,p),i=0..k);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*(,&%\"kG\"\"\"F&F&F&,&%\"pGF&F&F&!\"\"-%)binomial
G6$,(F(F&F%F&F&F&F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "co
nvert(\",GAMMA);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&%\"kG\"\"\"F&F
&F&,&%\"pGF&F&F&!\"\"-%&GAMMAG6#,(F(F&F%F&\"\"#F&F&-F+6#F'F)-F+6#,&F%F
&F.F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "convert(binomial
(p+k+1,p+1),GAMMA);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%&GAMMAG6#,(
%\"pG\"\"\"%\"kGF)\"\"#F)F)-F%6#,&F(F)F+F)!\"\"-F%6#,&F*F)F)F)F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(\"-\"\");" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
159 "Il s'agit d'une suite \"hyper-g\351om\351trique\" : on dispose d'
algorithmes pour \"calculer\" de telles sommes.\n\nCela dit, pour le 1
8, je n'ai pas r\351ussi \340 m'en sortir." }}}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 11 "Exercice 19" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "Q:=n->add((-1)^k*binomial(2^n-k,k),k=0..2^n):" }}{PARA 7 "" 1 "" 
{TEXT -1 42 "Warning, `k` in call to `add` is not local" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(Q(n),n=0..10);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6-\"\"\"\"\"!!\"\"F$F%F$F%F$F%F$F%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 41 "R:=p->add((-1)^k*binomial(p-k,k),k=0..p):
" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warning, `k` in call to `add` is not \+
local" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(R(p),p=0..30);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A\"\"\"F#\"\"!!\"\"F%F$F#F#F$F%F%F$
F#F#F$F%F%F$F#F#F$F%F%F$F#F#F$F%F%F$F#" }}}}}{MARK "4 2 0 0" 0 }
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