TITLE: Finding the coefficient of $x^{25}$ in $(1 + x^3 + x^8)^{10}$?
QUESTION [5 upvotes]: Find the coefficient of $x^{25}$ in $(1 + x^3 + x^8)^{10}$.

I've tried thinking of this combinatorially, but I couldn't get it to make sense. I've also tried applying some identities, only to lead to dead ends. Any hints?

REPLY [5 votes]: "The only way to form an $x^{25}$ term is to gather two $x^8$ and three $x^3$ . Since there are ${{10}\choose{2}} =45$ ways
to choose two $x^8$ from the $10$ multiplicands and $8$ ways to choose three  ${{8}\choose{3}}= 56$ ways to choose $x^3$ from the remaining $8$
multiplicands, the answer is $45×56 = 2520$." Same as asimut with slightly different wording.