Question

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

Short answer

The coefficient of$x^3$in the expansion of${\left(x-3\right)}^{10}$is$-262440$.

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 $x^3$in the expansion of ${\left(x-3\right)}^{10}$is:

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

$=\frac{10!}{7!\left(10-7\right)!}{\left(-3\right)}^7{x}^3$$=\frac{10\times 9\times 8\times 7!}{7!\times 3!}\left(-2187\right)x^3$

$=\left(120\right)\left(-2187\right)x^3$

$=-262440x^3$

So, the coefficient of $x^3$in the expansion of${\left(x-3\right)}^{10}=-262440$.