Question

Find the coefficient of $x^7$ in the expansion of${\left(x+2\right)}^9.$

Short answer

The coefficient of $x^7$is$144$.

Step-by-step solution

Given information:

Consider the expression ${\left(x+2\right)}^9$.

To find the coefficient of$x^7$.

Formulas :

Formula 1: To find a particular term in an expression:

${\left(x+a\right)}^n=\left(\begin{array}{c}n\\ n-j\end{array}\right)a^{\left(n-j\right)}{x}^j$

Formula 2: Binomial Coefficient:

$\left(\begin{array}{c}n\\ j\end{array}\right)=\frac{n!}{j!\times \left(n-j\right)!}$

Step 3:

On comparing ${\left(x+2\right)}^9$with ${\left(x+a\right)}^n$, we get,

$x=x,a=2,n=9.$

To find the coefficient of $x^7$,role="math" localid="1647343175949" $\Rightarrow j=7$

Substituting the values in the formula 1:

${\left(x+2\right)}^9=\left(\begin{array}{c}9\\ 9-7\end{array}\right){\left(2\right)}^{(9-7)}{x}^7$

Simplification:

${\left(x+2\right)}^9=\left(\begin{array}{c}9\\ 2\end{array}\right){\left(2\right)}^2{\left(x\right)}^7$

$=36\times 4x^7$ $\left[useformula2:\left(\begin{array}{c}9\\ 2\end{array}\right)=\frac{9!}{2!\left(9-2\right)!}=\frac{9!}{2!\times 7!}=\frac{9\times 8\times 7!}{\left(2\times 1\right)\times 7!}=\frac{9\times 8}{2}=36\right]$

role="math" localid="1647059684716" $=144x^7$