Question

What is the coefficient of${\mathit{x}}^{\mathbf{9}}\mathbf{\text{in}}\mathbf{(}\mathbf{2}\mathbf{-}\mathit{x}{\mathbf{)}}^{\mathbf{19}}$?

Short answer

The coefficient of$x^9\text{is}-94,595,072$

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 of n + 1terms 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$x^9\text{in}(2-x{)}^{19}=(2+(-x){)}^{19}$

$\begin{array}{r}n=19\\ j=9\end{array}$

Find the corresponding term

$\begin{array}{r}\left(\begin{array}{c}n\\ j\end{array}\right)2^{n-j}(-x{)}^j=\left(\begin{array}{c}19\\ 9\end{array}\right)2^{19-9}(-x{)}^9\\ =\frac{19!}{9!(19-9)!}{2}^{10}\left(-x^9\right)\\ =-92,378\cdot 1024x^9\\ =-94,595,072x^9\end{array}$

Thus, the coefficient of$x^9\text{is}-94,595,072$