<s>[INST] Find a coefficient of $x^{12}$ in $(x + x^{2} +x^{3} +x^{4})^{5}$ Find a coefficient of $x^{12}$ in $$(x + x^{2} +x^{3} +x^{4})^{5}$$
Using formula for the sum of a geometric series I have turned it into $$x^5(1-x^4)(1-x)^{-4} $$
So now I see that in $(1-x^4)(1-x)^{-4}$ we need to find coefficient of $x^7$ as we actually have $x^5$, am I right? 
If so I unfortunately still don't know what I should do as my next step to calculate this coefficient.
 [/INST] you can have a look at this post too : Coefficient of $x^{25}$ 
$$(x+x^2+x^3+x^4)^5=\sum\limits_{i+j+k+l=5} \binom{5}{i,j,k,l}x^{i+2j+3k+4l}$$
The coefficient $x^{12}$ is obtained when $i+2j+3k+4l=12$.
$\begin{array}{cccc|c}
i & j & k & l & \dfrac {5!}{i!j!k!l!}\\\hline
0 & 3 & 2 & 0 & 10\\
0 & 4 & 0 & 1 & 5\\
1 & 1 & 3 & 0 & 20\\\
1 & 2 & 1 & 1 & 60\\
2 & 0 & 2 & 1 & 30\\
2 & 1 & 0 & 2 & 30\\\hline
&&&&155\end{array}$
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