Question

What is the coefficient of${\mathit{x}}^{\mathbf{7}}\mathbf{\text{in}}\mathbf{(}\mathbf{1}\mathbf{+}\mathit{x}{\mathbf{)}}^{\mathbf{11}}$?

Short answer

The coefficient of$x^7\text{in}(1+x{)}^{11}$ 330.

Step-by-step solution

Use Binomial theorem:

Binomial theorem: binomial theorem, statement that for any positive integer n, the nth power of the sum of two numbers xand ymay be expressed as the sum ofterms of the form.

$\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}{\mathbf{)}}^{\mathbf{n}}\mathbf{=}\mathbf{\sum }_{\mathbf{j}\mathbf{=}\mathbf{0}}^{\mathbf{n}}\mathbf{ }\left(\begin{array}{l}n\\ j\end{array}\right){\mathit{x}}^{\mathbf{n}\mathbf{-}\mathbf{j}}{\mathit{y}}^{\mathbf{j}}$

The term$x^7$ (or equivalently $\left.1^4{x}^7\right)\text{in}(1+x{)}^{11}$

$\begin{array}{c}n=11\\ j=7\end{array}$

Find the coefficient of the given term

$\begin{array}{r}\left(\begin{array}{l}n\\ j\end{array}\right)=\left(\begin{array}{c}11\\ 7\end{array}\right)\\ =\frac{11!}{7!(11-7)!}\\ =\frac{11!}{7!4!}\\ =330\end{array}$

Hence, the coefficient of in is 330.