There are n identical cards of type A, n of type B, n of type C, and n of type D. There are 4 persons that each have to receive n cards. In how many ways can we distribute the cards ?

So the order in which they receive the cards does not matter. All that matters is how many cards of each type (A, B, C, or D) each person has. Can this be expressed with combinations depending on the parameter n ?

Question
Answer 1

See below for an idea on how to approach this answer:

I believe the answer to the question of methodology on doing this problem is that Combinations with identical items within the population (such as having #4n# cards with #n# number of types A, B, C, and D) falls outside the ability of the combination formula to calculate. Instead, according to Dr. Math at mathforum.org, you end up needing a couple of techniques: distributing objects into distinct cells, and the inclusion-exclusion principle.

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Answer 2

A counting program in C yields following results :

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#include

for(n=1;n<=20;n++) { numcomb = 0; for(i=0;i<=n;i++) for(j=0;j<=n-i;j++) for(k=0;k<=n-i-j;k++) { comb[numcomb][0] = i; comb[numcomb][1] = j; comb[numcomb][2] = k; comb[numcomb][3] = n-i-j-k; numcomb++; } count = 0; for(i=0;in) br = 1; if (!br) { for(k=0;kn) br2=1; if (!br2) { count++; } } } } } printf("\nCount for n=%d : %ld.", n, count); } printf("\n"); return(0); }

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