<s>[INST] Finding a Particular Coefficient Using Generating Functions I have a homework question to solve the number of ways to choose 25 ice creams of a selection of 6 types of ice creams and there are only 7 of each ice cream type.
The question requires the use of generating functions and I have gotten to this point:
$$({{x}^{48}}-6{{x}^{40}}+15{{x}^{32}}-20{{x}^{24}}+15{{x}^{16}}-6{{x}^{8}}+1)\cdot {{\left( \frac{1}{1-x} \right)}^{6}}$$
But know I need to find the coefficient of $x^{25}$ and I don't know how to continue. 
Can anyone please help?
Thanks a lot,
 [/INST] It might be useful (or not) to note that the first factor is $(z^8 - 1)^6$. Try:
$\begin{align*}
[x^{25}] (x^{48}-6 x^{40} + 15 x^{32} - 20 x^{24} + 15 x^{16} -6 x^{8} + 1)
           \cdot \left( \frac{1}{1 - x} \right)^6
  &= [x^{25}] (1 - 6 x^8 + 15 x^{16} - 20 x^{24})
                \cdot \sum_{k \ge 0} (-1)^k \binom{-6}{k} x^k \\
  &= ([x^{25}] - 6 [x^{17}] + 15 [x^9] - 20 [z])
        \sum_{k \ge 0} \binom{k + 6 - 1}{6 - 1} z^k \\
  &= \binom{25 + 5}{5} - 6 \binom{17 + 5}{5} + 15 \binom{9 + 5}{5} 
        - 20 \binom{1 + 5}{5} \\
  &= 14412
\end{align*}$
</s>