<s>[INST] Finding the coefficient in an expansion? I'm reviewing for my combinatorics final and have completely forgotten how to find the coefficient in the expansion of a polynomial. Here's an example I'm struggling with

Find the coefficient of $x^{11}$ in the expansion of
  $$(x+x^2+x^3+x^4+x^5)^7(1+x+x^2+x^3+\dots)^4$$

So first I know how to rewrite it as
$$x^7(1-x^5)^7\frac1{(1-x)^{11}}$$
But I have no idea how to find the coefficients from here.
 [/INST] Hint Pulling out the factor of $x^7$ we see that this is the same as finding the coefficient of $x^4$ in the expansion of $$(1 + x + x^2 + x^3 + x^4)^7 (1 + x + x^2 + x^3 + x^4 + \cdots)^4 .$$ Since no term of degree $\geq 5$ can contribute, this is the same as the coefficient of $x^4$ in
$$(1 + x + x^2 + x^3 + x^4)^7 (1 + x + x^2 + x^3 + x^4)^4 = (1 + x + x^2 + x^3 + x^4)^{11} .$$
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