[obm-l] Re: [obm-l] Re: [obm-l] Análise Combinatória

["Rafael" <cyberhelp@xxxxxxxxxx>]

Oi, Claudio:

Se eu não errei ao digitar as expressões indicadas por você, o MathCAD
responde:

1+y^3+y+48*x^10*y^3+432*x^25*y^21+1068*x^23*y^20+1336*x^15*y^17+228*x^15*y^1
8+4*x^15*y^19+16*x^7*y^2+1776*x^27*y^17+1776*x^27*y^18+1776*x^27*y^19+1648*x
^27*y^20+2308*x^26*y^16+2308*x^26*y^17+2308*x^26*y^18+2292*x^26*y^19+1836*x^
26*y^20+692*x^26*y^21+28*x^11*y^17+332*x^11*y^16+719*x^12*y^16+113*x^12*y^17
+x^12*y^18+1290*x^28*y^16+1290*x^28*y^17+1290*x^28*y^18+1290*x^28*y^19+1266*
x^28*y^20+908*x^28*y^21+y^2+y^10+1483*x^24*y^20+4560*x^19*y^13+4560*x^19*y^1
4+4560*x^19*y^15+4560*x^19*y^16+4272*x^19*y^17+233*x^12*y^4+24*x^3*y^3+10*x^
2*y^2+4*x*y+114*x^6*y^3+26*x^4*y^2+4*x^2*y+22*x^6*y^2+y^9+4*x^3*y+2664*x^19*
y^18+4250*x^22*y^14+4250*x^22*y^15+4250*x^22*y^16+4250*x^22*y^17+4018*x^22*y
^18+2522*x^22*y^19+2856*x^25*y^16+2856*x^25*y^17+2856*x^25*y^18+2744*x^25*y^
19+1808*x^25*y^20+1920*x^21*y^19+3876*x^23*y^15+3876*x^23*y^16+3876*x^23*y^1
7+3812*x^23*y^18+2932*x^23*y^19+3395*x^24*y^18+3007*x^24*y^19+1292*x^13*y^16
+316*x^13*y^17+4488*x^21*y^17+3872*x^21*y^18+4*x^12*y^3+6*x^8*y^2+y^8+40*x^3
4*y^15+40*x^34*y^16+40*x^34*y^17+16*x^35*y^15+16*x^35*y^16+16*x^35*y^17+16*x
^35*y^18+344*x^31*y^14+344*x^31*y^15+4*x^36*y^16+4*x^36*y^17+4*x^36*y^18+4*x
^36*y^19+194*x^32*y^15+896*x^29*y^15+580*x^30*y^14+580*x^30*y^15+580*x^30*y^
16+347*x^8*y^4+51*x^4*y^4+896*x^29*y^14+128*x^7*y^3+32*x^5*y^2+4*x^24*y^7+16
*x^21*y^6+24*x^18*y^5+16*x^15*y^4+102*x^20*y^6+88*x^17*y^5+864*x^27*y^21+128
*x^27*y^22+92*x^9*y^3+4*x^4*y+52*x^14*y^4+24*x^11*y^3+236*x^16*y^5+16*x^13*y
^18+128*x^13*y^4+y^4+y^5+52*x^26*y^22+3100*x^16*y^7+50*x^4*y^3+356*x^30*y^22
+16*x^25*y^22+896*x^29*y^18+896*x^29*y^19+896*x^29*y^20+784*x^29*y^21+580*x^
30*y^18+580*x^30*y^19+580*x^30*y^20+564*x^30*y^21+3379*x^20*y^8+948*x^10*y^6
+632*x^9*y^5+284*x^7*y^6+170*x^6*y^5+96*x^5*y^4+284*x^7*y^5+445*x^8*y^6+20*x
^3*y^2+80*x^5*y^3+166*x^6*y^4+634*x^22*y^20+28*x^22*y^21+237*x^24*y^21+x^24*
y^22+92*x^23*y^21+952*x^10*y^7+2516*x^18*y^7+1488*x^16*y^6+696*x^14*y^5+2956
*x^17*y^7+1232*x^20*y^19+124*x^20*y^20+696*x^19*y^19+32*x^19*y^20+308*x^21*y
^20+4*x^21*y^21+1104*x^17*y^18+112*x^17*y^19+1840*x^15*y^6+549*x^16*y^18+28*
x^16*y^19+1854*x^18*y^18+306*x^18*y^19+6*x^18*y^20+892*x^13*y^5+732*x^14*y^1
7+68*x^14*y^18+328*x^11*y^4+96*x^33*y^18+1948*x^14*y^6+997*x^12*y^5+40*x^34*
y^18+408*x^10*y^4+118*x^8*y^3+972*x^11*y^5+96*x^33*y^17+344*x^31*y^16+344*x^
31*y^17+194*x^32*y^16+194*x^32*y^17+194*x^32*y^18+396*x^9*y^4+896*x^29*y^17+
580*x^30*y^17+96*x^33*y^15+96*x^33*y^16+896*x^29*y^16+664*x^9*y^7+445*x^8*y^
7+445*x^8*y^8+2168*x^13*y^7+16*x^34*y^24+80*x^33*y^23+190*x^32*y^22+94*x^32*
y^23+6*x^32*y^24+48*x^30*y^23+280*x^31*y^22+88*x^31*y^23+4196*x^21*y^9+2808*
x^23*y^9+3162*x^24*y^10+96*x^5*y^6+96*x^5*y^5+51*x^4*y^8+284*x^7*y^7+51*x^4*
y^7+3864*x^19*y^8+96*x^33*y^22+312*x^29*y^22+24*x^29*y^23+3616*x^22*y^9+16*x
^35*y^21+16*x^35*y^22+16*x^35*y^23+16*x^35*y^24+96*x^33*y^19+96*x^33*y^20+96
*x^33*y^21+40*x^34*y^19+40*x^34*y^20+40*x^34*y^21+40*x^34*y^22+40*x^34*y^23+
16*x^35*y^19+16*x^35*y^20+2424*x^25*y^10+1058*x^28*y^11+912*x^27*y^10+170*x^
6*y^7+51*x^4*y^5+51*x^4*y^6+4*x^36*y^20+4*x^36*y^21+4*x^36*y^22+4*x^36*y^23+
4*x^36*y^24+4*x^36*y^25+344*x^31*y^18+344*x^31*y^19+344*x^31*y^20+344*x^31*y
^21+194*x^32*y^20+194*x^32*y^21+194*x^32*y^19+4*x*y^5+170*x^6*y^6+4*x*y^2+4*
x*y^3+4*x*y^4+232*x^28*y^22+4*x^28*y^23+24*x^3*y^6+10*x^2*y^3+10*x^2*y^4+10*
x^2*y^5+10*x^2*y^6+24*x^3*y^7+24*x^3*y^4+24*x^3*y^5+1716*x^12*y^8+1304*x^11*
y^7+1603*x^12*y^6+632*x^21*y^7+2256*x^26*y^11+1776*x^27*y^12+288*x^19*y^6+37
84*x^23*y^10+3176*x^15*y^9+1716*x^12*y^9+2840*x^25*y^11+2680*x^14*y^10+2184*
x^13*y^8+1304*x^11*y^8+952*x^10*y^11+3176*x^15*y^10+4060*x^17*y^11+2680*x^14
*y^9+4060*x^17*y^10+1715*x^12*y^7+2680*x^14*y^8+3649*x^16*y^9+4370*x^18*y^11
+4370*x^18*y^10+1916*x^24*y^9+1728*x^22*y^8+1186*x^20*y^7+624*x^18*y^6+3172*
x^15*y^8+284*x^7*y^9+2584*x^21*y^8+445*x^8*y^9+445*x^8*y^10+4060*x^17*y^9+17
0*x^6*y^8+170*x^6*y^9+32*x^9*y^16+96*x^5*y^8+284*x^7*y^8+96*x^5*y^7+4528*x^1
9*y^9+664*x^9*y^8+664*x^9*y^9+4560*x^19*y^10+4504*x^21*y^11+4364*x^18*y^9+95
2*x^10*y^8+952*x^10*y^9+952*x^10*y^10+1896*x^19*y^7+4611*x^20*y^10+4611*x^20
*y^11+4611*x^20*y^12+4500*x^21*y^10+1064*x^17*y^6+4250*x^22*y^11+3399*x^24*y
^12+3649*x^16*y^10+3649*x^16*y^11+3876*x^23*y^11+432*x^15*y^5+4222*x^22*y^10
+3948*x^17*y^8+4487*x^20*y^9+4064*x^18*y^8+16*x^33*y^24+584*x^29*y^11+532*x^
30*y^12+2856*x^25*y^12+2856*x^25*y^15+2184*x^13*y^12+3176*x^15*y^12+3176*x^1
5*y^13+3176*x^15*y^11+3876*x^23*y^14+2184*x^13*y^9+1776*x^27*y^15+1776*x^27*
y^16+2308*x^26*y^13+2308*x^26*y^14+2308*x^26*y^15+382*x^28*y^10+472*x^26*y^9
+1304*x^11*y^11+1304*x^11*y^10+1776*x^27*y^14+1716*x^12*y^10+1716*x^12*y^11+
1716*x^12*y^12+1776*x^27*y^13+1290*x^28*y^14+1290*x^28*y^15+2856*x^25*y^13+2
856*x^25*y^14+4250*x^22*y^12+4250*x^22*y^13+4560*x^19*y^11+4560*x^19*y^12+33
99*x^24*y^13+3399*x^24*y^14+392*x^24*y^8+3398*x^24*y^11+3876*x^23*y^12+3876*
x^23*y^13+4504*x^21*y^13+232*x^22*y^7+2308*x^26*y^12+2184*x^13*y^10+2184*x^1
3*y^11+4504*x^21*y^12+1048*x^25*y^9+944*x^23*y^8+1290*x^28*y^13+1304*x^11*y^
9+268*x^7*y^13+96*x^5*y^9+96*x^5*y^10+4*x^32*y^11+16*x^30*y^10+y^6+x^4*y^14+
25*x^4*y^13+327*x^8*y^14+24*x^28*y^9+4*x^36*y^15+64*x^31*y^11+16*x^26*y^8+44
5*x^8*y^11+445*x^8*y^12+112*x^29*y^10+128*x^27*y^9+112*x^25*y^8+24*x^34*y^13
+16*x^35*y^14+64*x^23*y^7+16*x^33*y^12+896*x^29*y^13+51*x^4*y^9+51*x^4*y^11+
47*x^4*y^12+51*x^4*y^10+170*x^6*y^12+148*x^6*y^13+4*x*y^11+170*x^6*y^10+170*
x^6*y^11+4*x*y^8+4*x*y^9+4*x*y^10+10*x^2*y^10+10*x^2*y^11+6*x^2*y^12+4*x*y^6
+4*x*y^7+4*x^3*y^13+10*x^2*y^7+10*x^2*y^8+10*x^2*y^9+24*x^3*y^8+24*x^3*y^9+2
4*x^3*y^10+24*x^3*y^11+20*x^3*y^12+1716*x^12*y^13+1712*x^12*y^14+284*x^7*y^1
0+284*x^7*y^11+284*x^7*y^12+952*x^10*y^13+952*x^10*y^12+1286*x^28*y^12+1648*
x^27*y^11+1616*x^26*y^10+580*x^30*y^13+872*x^29*y^12+344*x^31*y^13+194*x^32*
y^14+3649*x^16*y^12+3649*x^16*y^13+3649*x^16*y^14+1304*x^11*y^12+1304*x^11*y
^13+2680*x^14*y^12+2680*x^14*y^13+2680*x^14*y^14+2680*x^14*y^11+664*x^9*y^12
+664*x^9*y^13+664*x^9*y^10+664*x^9*y^11+4060*x^17*y^14+4060*x^17*y^15+4060*x
^17*y^12+4060*x^17*y^13+4611*x^20*y^14+4611*x^20*y^16+4611*x^20*y^15+3176*x^
15*y^14+4611*x^20*y^13+4370*x^18*y^12+4370*x^18*y^13+4370*x^18*y^14+4370*x^1
8*y^15+4504*x^21*y^14+4504*x^21*y^15+4504*x^21*y^16+3399*x^24*y^15+3399*x^24
*y^16+3399*x^24*y^17+96*x^5*y^11+96*x^5*y^12+3425*x^20*y^18+3413*x^16*y^16+2
161*x^16*y^17+2184*x^13*y^13+2612*x^14*y^7+2184*x^13*y^14+439*x^8*y^13+3649*
x^16*y^15+1868*x^13*y^6+4346*x^18*y^16+3746*x^18*y^17+4*x^10*y^17+64*x^5*y^1
3+16*x^5*y^14+6*x^8*y^16+98*x^8*y^15+28*x^7*y^15+156*x^7*y^14+904*x^10*y^14+
544*x^10*y^15+128*x^10*y^16+1276*x^11*y^6+824*x^10*y^5+56*x^6*y^14+4*x^6*y^1
5+268*x^9*y^15+572*x^9*y^14+4509*x^20*y^17+3972*x^17*y^16+2996*x^17*y^17+262
8*x^14*y^15+100*x^32*y^12+80*x^33*y^13+1984*x^14*y^16+1280*x^11*y^14+2056*x^
13*y^15+2744*x^15*y^16+3160*x^15*y^15+40*x^34*y^14+188*x^32*y^13+224*x^30*y^
11+2948*x^15*y^7+y^7+439*x^8*y^5+256*x^7*y^4+3621*x^16*y^8+1483*x^12*y^15+96
*x^33*y^14+256*x^31*y^12+664*x^9*y^6+976*x^11*y^15

Procurando o coeficiente de x^12*y^13 encontramos 1716.

Boas férias!


[]s,

Rafael



----- Original Message -----
From: claudio.buffara
To: obm-l
Sent: Monday, July 05, 2004 3:52 PM
Subject: [obm-l] Re: [obm-l] Análise Combinatória


Oi, Carlos:

Eh que o seu enunciado foi um pouco longo, o que pode ter feito com que a
maioria das pessoas desistisse de le-lo ateh o fim.

O baralho tem:
4 A: 4 pontos cada
4 K: 3 pontos cada
4 Q: 2 pontos cada
4 J: 1 ponto cada
36 numeros: 0 pontos cada.

Voce quer saber o numero de maos de 13 cartas cuja soma eh 12 pontos.

Isso eh igual ao numero de solucoes do sistema:
4*(a1+a2+a3+a4) + 3*(k1+k2+k3+k4) + 2*(q1+q2+q3+q4) + (j1+j2+j3+j4) = 12;

a1+a2+a3+a4+k1+k2+k3+k4+q1+q2+q3+q4+j1+j2+j3+j4+n1+n2+ ... +n36 = 13,

onde o universo das 52 variaveis eh igual a {0,1}.

Isso eh igual ao coeficiente de x^12*y^13 na expansao de:
(1+4x^4y+6x^8y^2+4x^12y^3)*
(1+4x^3y+6x^6y^2+4x^9y^3+x^12y^4)*
(1+4x^2y+6x^4y^2+4x^6y^3+x^8y^4)*
(1+4xy+6x^2y^2+4x^3y^3+x^4y^4)*(1+y+y^2+y^3+y^4+y^5+y^6+y^7+y^8+y^9+y^10)

Repare que x controla a soma dos pontos e y o numero de cartas.

Infelizmente, eu estou de ferias no Rio de Janeiro, sem acesso a qualquer
tipo de software matematico, de modo que nao vou conseguir dar a resposta
numerica que voce deseja (fazer na mao nem pensar!)

[]s,
Claudio.


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