<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] I should mention, this answer assumes the dice are distinct.  (e.g. you have a red die, a blue die, a green die, etc... or a "first die" a "second die" a "third die", etc...)
We know that each result on the die is minimum 1, so let us ask a similar question instead to simplify the arithmetic involved:  What is the coefficient of $x^{15}$ in the expansion of $(1+x+x^2+\dots+x^5)^{10}$.
Find how many ways you can add to 15 using 1's, 2's, 3's, etc in decreasing order using at most 10 positive integers less than or equal to 5.
$$\begin{array}{l}5+5+5\\
5+5+4+1\\
5+5+3+2\\
5+5+3+1+1\\
5+5+2+2+1\\
5+5+2+1+1+1\\
5+5+1+1+1+1+1\\
5+4+4+2\\
\color{red}{5+4+4+1+1}\\
5+4+3+3\\
5+4+3+2+1\\
5+4+3+1+1+1\\
5+4+2+1+1+1+1\\
5+4+1+1+1+1+1+1\\
\vdots\\
2+2+2+2+2+1+1+1+1+1\end{array}$$
For each of these, you can calculate their specific contribution.  The line colored red for example occurs when you use one occurrence of $x^5$, two occurrences of $x^4$, two occurrences of $x^1$, and the remaining five occurrences as $x^0$.  If you were to reimagine this as finding the coefficient of $A^1B^2C^2D^5$ in the expansion of $(A+B+C+D+E+F)^{10}$, it will be $\binom{10}{1,2,2,5}=\frac{10!}{1!2!2!5!}$.
I.e., the red line corresponds to rolling one 6, two 5s, two 2s, and five 1s.
Find the contribution of each case separately and add them together.

Of course, this is a very tedious process to do by hand, and so it is much easier to simply ask a calculator to expand $(1+x+x^2+x^3+x^4+x^5)^{10}$ and find the coefficient that way.
Wolfram shows that it is $831204$

If you are curious in the number of ways you can do so where the dice are indistinct, just count how many ways are in the list above.
</s>