Mathematics - Binomial Theorem Question with Solution | TestHub

The expression inside the parenthesis is the Taylor series expansion of $e^x$ truncated to the $x^5$ term. We need to find the coefficient of $x^6$ in ${\left(1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}\right)}^2$. This is obtained by multiplying terms such that the powers add up to 6:

$$\frac{1}{1!5!}+\frac{1}{2!4!}+\frac{1}{3!3!}+\frac{1}{4!2!}+\frac{1}{5!1!}=\frac{2}{120}+\frac{2}{48}+\frac{1}{36}=\frac{1}{60}+\frac{1}{24}+\frac{1}{36}=\frac{6+15+10}{360}=\frac{31}{360}$$

Thus, the coefficient of $x^6$ is $\frac{31}{360}$.