How many times do I have to expand?

Question

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!

Answer 1

\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

$$2\begin{pmatrix} 10 \\ 3\end{pmatrix} -\begin{pmatrix} 10 \\ 4\end{pmatrix}$$

Answer 2

$$(2p-q)(p+q)^{10}=2p\sum_{i=0}^{10}\binom{10}{i} p^{i}q^{10-i} -q\sum_{i=0}^{10}\binom{10}{i}p^{i}q^{10-i}$$ $$=\sum_{i=0}^{10}2\binom{10}{i} p^{i+1}q^{10-i} -\sum_{i=0}^{10}\binom{10}{i}p^{i}q^{11-i}$$ 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 $p^4q^7$

Answer 3

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