Question

The coefficient of $x^n$ in the expansion of $(1-x)e^x$ is

• A. $\frac{1}{n!}$
• B. $\frac{1}{(n+1)!}$
• C. $\frac{1-n}{n!}$
• D. $\frac{n-1}{n!}$

Answer 1

The correct option is C $\frac{1-n}{n!}$

$(1-x)e^x=e^x-x{e}^x=[1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots ]-x[1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots ]$
Coefficient of $x^n$ will be
$=\frac{1}{n!}-\frac{1}{(n-1)!}=\frac{1}{n\times (n-1)!}-\frac{1}{(n-1)!}=\frac{1-n}{n!}$