The coefficient of $x^3$ in $(1+x)^3 \cdot (2+x^2)^{10}$ Find the coefficient of $x^3$ in the expansion $(1+x)^3 \cdot (2+x^2)^{10}$.
I did the first part, which is expanding the second equation at $x^3$ and I got: $\binom {10} 3 \cdot 2^7 \cdot (x^2)^3 = 15360 (x^2)^3$, but I can't figure out what to do from here.

It's also convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ in an algebraic expression.

We obtain
  \begin{align*}
[x^3](1+x)^3(2+x^2)^{10}&=[x^3]\left(\sum_{k=0}^{10}\binom{10}{k}x^{2k}2^{10-k}\right)(1+x)^3\\
&=[x^3]\left(\binom{10}{0}2^{10}x^0+\binom{10}{1}2^9x^2\right)(1+x)^3\tag{1}\\
&=\left(2^{10}[x^3]+10\cdot2^9[x^1]\right)\sum_{k=0}^{3}\binom{3}{x}x^k\tag{2}\\
&=\left(2^{10}\binom{3}{3}+10\cdot2^9\binom{3}{1}\right)\\
&=2^{10}+30\cdot 2^9\\
&=16384
\end{align*}

Comment:


*

*In (1) we ignore all summands in the first sum with powers of $x$ greater than 3

*In (2) we use the rule $[x^{k+l}]f(x)=[x^k]x^{-l}f(x), k,l\geq 0$ and the linearity of the coefficient of operator.
