How many times do I have to expand? For the binomial expansion:
If a question asks you to find the co-efficient of $p^4q^7$ in the expansion $(2p-q)$$(p+q)^{10}$
I would firstly expand out $(p+q)$ correct?  But how do I know how many times I have to expand it out?  
Thanks!
 A: \begin{align}
(2p-q)(p+q)^{10}&=2p\left(\sum_{i=0}^{10}\begin{pmatrix} 10 \\ i\end{pmatrix} p^{i}q^{10-i}\right) -q\left(\sum_{i=0}^{10}\begin{pmatrix} 10 \\ i\end{pmatrix} p^{i}q^{10-i}\right) \\
&=2\left(\sum_{i=0}^{10}\begin{pmatrix} 10 \\ i\end{pmatrix} p^{i+1}q^{10-i}\right) -\left(\sum_{i=0}^{10}\begin{pmatrix} 10 \\ i\end{pmatrix} p^{i}q^{11-i}\right)
\end{align}
For the first term, $p^4q^7$ term appears when $i=3$ and for the second term, $p^4q^7$ term appears when $i=4$.
Hence, the quantity of interest is 
$$2\begin{pmatrix} 10 \\ 3\end{pmatrix} -\begin{pmatrix} 10 \\ 4\end{pmatrix}$$
A: $$(2p-q)(p+q)^{10}=2p\sum_{i=0}^{10}\binom{10}{i} p^{i}q^{10-i} -q\sum_{i=0}^{10}\binom{10}{i}p^{i}q^{10-i}$$
$$=\sum_{i=0}^{10}2\binom{10}{i} p^{i+1}q^{10-i} -\sum_{i=0}^{10}\binom{10}{i}p^{i}q^{11-i}$$
in first sum for $i=3$ and in second sum for $i=4$ we get that
$$2\binom{10}{3}-\binom{10}{4}$$
is coefficient of $p^4q^7$
A: Using the Coefficient Extraction Operator,
$$
\begin{align}
\left[p^4q^7\right](2p-q)(p+q)^{10}
&=2\left[p^4q^7\right]p(p+q)^{10}-\left[p^4q^7\right]q(p+q)^{10}\tag{1}\\[6pt]
&=2\left[p^3q^7\right](p+q)^{10}-\left[p^4q^6\right](p+q)^{10}\tag{2}\\
&=2\binom{10}{3}-\binom{10}{4}\tag{3}
\end{align}
$$
Explanation:
$(1)$: the Coefficient Extraction Operator is linear
$(2)$: $\left[x^n\right]x\,f(x)=\left[x^{n-1}\right]f(x)$
$(3)$: apply the Binomial Theorem to evaluate the Coefficient Extraction Operator
