The coefficient of  xnin the expansion of 1+11!x+12!x2…+1n!xn2

# The coefficient of  ${x}^{n}$ ${\left(1+\frac{1}{1!}x+\frac{1}{2!}{x}^{2}\dots +\frac{1}{n!}{x}^{n}\right)}^{2}$

1. A

$\frac{{2}^{n}}{n!}$

2. B

$\frac{{2}^{n}}{n}$

3. C

$n!$

4. D

$\frac{1}{n!}$

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### Solution:

Coefficient of ${x}^{n}$ in

${\left(1+\frac{1}{1!}x+\frac{1}{2!}{x}^{2}+\dots +\frac{1}{n!}{x}^{n}\right)}^{2}$

= coefficient of  ${x}^{n}$ is

${\left(1+\frac{1}{1!}x+\frac{1}{2!}{x}^{2}+\dots +\frac{1}{n!}{x}^{n}+\frac{1}{\left(n+1\right)!}{x}^{n+1}\dots \right)}^{2}$

= coefficient of ${x}^{n}$ in  =   ${e}^{2x}=\left(1+2x+\frac{{2}^{2}{x}^{2}}{2!}+\frac{{2}^{3}{x}^{3}}{3!}\cdots \right)$

$=\frac{{2}^{n}}{n!}$

Alternate Solution

Coefficient of  ${x}^{n}$ in

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