<s>[INST] Finding the coefficient of expansion Question:
Find the coefficient of $x^{11}$ in the expansion of:$$(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$$
The traditional way of doing this, as far as I know, is to first find the coefficient of each term that has $x^{11}$, and then sum it. However, with three individual terms in multiplication, this would be a very tedious job. Is there a shorter way to approach such problems?
 [/INST] To calculate the coefficient by hand is somewhat cumbersome. Here is a notation which helps to make it better manageable. It is convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ of a series.

We obtain by successively applying the binomial theorem
  \begin{align*}
[x^{11}]&(1+x^2)^4(1+x^3)^7(1+x^4)^{12}\\
&=[x^{11}]\sum_{j=0}^{12}\binom{12}{j}x^{4j}(1+x^2)^4(1+x^3)^7\tag{1}\\
&=\sum_{j=0}^{2}\binom{12}{j}[x^{11-4j}](1+x^2)^4(1+x^3)^7\tag{2}\\
&=\sum_{j=0}^{2}\binom{12}{j}[x^{11-4j}]\sum_{k=0}^7\binom{7}{k}x^{3k}(1+x^2)^4\tag{3}\\
&=\sum_{j=0}^{2}\sum_{k=0}^3\binom{12}{j}\binom{7}{k}[x^{11-4j-3k}](1+x^2)^4\qquad\qquad\quad 4j+3k\leq 11\tag{4}\\
&=\sum_{j=0}^{2}\sum_{k=0}^3\binom{12}{j}\binom{7}{k}[x^{11-4j-3k}]
\sum_{l=0}^{4}\binom{4}{l}x^{2l}\qquad\qquad 4j+3k\leq 11\tag{5}\\
&=\sum_{j=0}^{2}\sum_{k=0}^3\sum_{l=0}^{4}
\binom{12}{j}\binom{7}{k}\binom{4}{l}\qquad\qquad\qquad\qquad 4j+3k+2l= 11\tag{6}\\
\end{align*}

Comment:


*

*In  (1) we start expanding the binomial with the greatest power $x^4$ which keeps the number of summands small as we will see in the next step.

*In (2) we use the linearity of the coefficient of operator and apply the rule $[x^{m-n}]=[x^m]x^n$. Note that since the power of $x$ is non negative, the upper limit of the index $j$ is now $2$.

*In (3) we expand the binomial $(1+x^3)^7$. 

*In (4) we do a similar job as in (2) and keep in mind, that  the power of $x$ is non-negative by stating $4j+3k\leq 11$.

*In (5) we expand the last binomial

*In (6) we apply the coefficient of operator the last time and obtain a summation formula for the coefficient of $x^{11}$.

The final step is to do some bookkeeping and look according to (6) for all triples $(j,k,l)$ with
\begin{align*}
\begin{matrix}
0\leq j \leq 2\\
0\leq k \leq 3\\
 0\leq l \leq 4\\
\end{matrix}
\qquad\quad\text{and}\quad\qquad 4j+3k+2l= 11
\end{align*}
We find
  \begin{array}{rrrrr}
  j&k&l&\binom{12}{j}\binom{7}{k}\binom{4}{l}&\sum\\
\hline
  2&1&0&462&462\\
  1&1&2&504&966\\
  0&1&4&7&973\\
  0&3&1&140&\color{blue}{1113}\\
  \end{array}
resulting finally in
\begin{align*}
[x^{11}](1+x^2)^4(1+x^3)^7(1+x^4)^{12}=1113
\end{align*}
in accordance with the result of Wolfram Alpha.

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