Number of ways distribute 12 identical action figures to 5 children

Question

Need a little help with this problem.

Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most three action figures.

So far I have that we are looking for the coefficient of $x^{12}$ and the generating function is $G(x) = (1 + x + x^2 + x^3)^{5}$ so this is equal to the form $G(x) = 1/(1-x)^n$ which is equal to $(1-x)^{-n}$ and then I'm trying to use the formula $C(n+k-1,k)x^k$ and using $k=12$ and $n=5$ to come out with $C(16,12)$ but I'm not sure if that is correct.

I don't know if I'm doing this correctly or messing up a step along the way. Any help greatly appreciated.

There is no way to distribute 12 things to 5 kids that will not cause problems. — Mariano Suárez-Álvarez, Jul 22 '14 at 21:58
thank you, that made me chuckle, but alas this problem does not take into consideration hurt feelings. — MannfromReno, Jul 22 '14 at 21:59
I believe a generalization of this is answered in http://math.stackexchange.com/q/89417/139123 — David K, May 16 '15 at 04:46

score 2 · Accepted Answer · answered Jul 22 '14 at 23:48

2

You have $\displaystyle G(x)=(1+x+x^2+x^3)^5=\big(\frac{1-x^4}{1-x}\big)^5=(1-x^4)^5(1-x)^{-5}$

$\displaystyle=\big(1-5x^4+\binom{5}{2}x^8-\binom{5}{3}x^{12}+\cdots\big)\big(\sum_{k=0}^{\infty}\binom{k+4}{4}x^4\big)$,

so now you just have to find the coefficient of $x^{12}$ in this expression.

answered Jul 22 '14 at 23:48

user84413

27,211

Thank you, I see where I made my mistake by using an infinte series rather than the correct series going to n – MannfromReno Jul 23 '14 at 11:48

Number of ways distribute 12 identical action figures to 5 children

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