Question

The coefficient of$x^6$ in the expansion of${\left(x+3\right)}^{10}$

Short answer

The coefficient of$x^6$in the expansion of${\left(x+3\right)}^{10}$is$17010$.

Step-by-step solution

Step 1. Definition

Based on the expansion of ${\left(x+a\right)}^n$, the term containing$x^j$is$\left(\begin{array}{c}n\\ n-j\end{array}\right)a^{n-j}{x}^j$.

Step 2. Explanation

The term containing $x^6$in the expansion of ${\left(x+3\right)}^{10}$is:

localid="1647190158816" $=\left(\begin{array}{c}10\\ 10-6\end{array}\right){\left(3\right)}^{10-6}{x}^6$

$=\frac{10!}{4!\left(10-4\right)!}{\left(3\right)}^4{x}^6$

$=\frac{10\times 9\times 8\times 7\times 6!}{4!\times 6!}\left(81\right)x^6$

$=\left(210\right)\left(81\right)x^6$

$=17010x^6$

So, the coefficient of $x^6$in the expansion of ${\left(x+3\right)}^{10}=17010$.