<s>[INST] Expansion of Generating Functions If you roll $10$ dice, how many ways can you get a total sum of top faces of $25$?
I understand how to write the generating function of $(x+x^2+ \dots +x^6)^{10}$ and the fact that you need to find the coefficient of $x^{25}$, but how do you do this? The conventional binomial theorem only works for binomial expansion....
 [/INST] Write $x+x^2+\cdots+x^6$ as $(1-x^7)(1-x)^{-1}$. Then
$$\begin{align}
[x^{25}]&(x+x^2+\cdots+x^6)^{10}\\
&=[x^{25}](x-x^7)^{10}(1-x)^{-10}\\
&=[x^{15}](1-x^6)^{10}(1-x)^{-10}\\
&=[x^{15}]\sum_{r\ge0}\binom{10}r(-x^6)^r
    \sum_{s\ge0}\binom{-10}s(-x)^s\\
&=\sum_{r\ge0}(-1)^r\binom{10}r\binom{10+s-1}{10-1}\quad\textrm{with $s=15-6r$}\\
&=\sum_{r=0}^2(-1)^r\binom{10}r\binom{24-6r}{9}\quad\textrm{noting $15-6r\ge0$}\\
&=\binom{10}0\binom{24}9 -\binom{10}1\binom{18}9 +\binom{10}2\binom{12}9\\
&=831204
\end{align}$$
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