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.
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.