Question

Find the coefficient of${\mathit{x}}^{\mathbf{5}}{\mathit{y}}^{\mathbf{8}}\mathbf{\text{in}}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}{\mathbf{)}}^{\mathbf{13}}$.

Short answer

$\text{The coefficient of}{x}^5{y}^8\text{in}(x+y{)}^{13}\text{is}1287\text{.}$

Step-by-step solution

Use Binomial theorem:

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

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

The term$x^5{y}^8\text{in}(x+y{)}^{13}$

$\begin{array}{r}n=13\\ j=8\end{array}$

Find the coefficient of the given term

$\begin{array}{r}\left(\begin{array}{c}n\\ j\end{array}\right)=\left(\begin{array}{c}13\\ 8\end{array}\right)\\ =\frac{13!}{8!(13-8)!}!\\ =\frac{13!}{8!5!}\\ =1287\end{array}$

Hence, the coefficient of$x^5{y}^8\text{in}(x+y{)}^{13}$ is 1287.