I know I can use generating functions. Each of the die has a generating function $x+x^2+x^3+x^4+x^5+x^6$, and so I need to find the coefficient of $x^9$ in the generating function of their sum, $(x+x^2+x^3+x^4+x^5+x^6)^3$. I am not sure how to do this, however, short of expanding it all out.
(I'm not just trying to get the answer but also the method behind it. Thanks a lot.)