The coefficient of xn in 1+x1!+x22!+⋯+xnn!2

1+x1!+x22!+⋯+xnn!2=1+x1!+x22!+⋯+xnn!1+x1!+x22!+⋯+xn−1(n−1)!+xnn! coefficient of xn=1⋅1n!+11!⋅1(n−1)!+12!⋅1(n−2)!+⋯+1n!1=1n!n!n!+n!(n−1)!1!+n!(n−2)!2!+⋯+n!n!=1n! nC0+nC1+nC2+⋯+nCn=2nn!

The coefficient of xn in 1+x1!+x22!+⋯+xnn!2

1+x1!+x22!+⋯+xnn!2=1+x1!+x22!+⋯+xnn!1+x1!+x22!+⋯+xn−1(n−1)!+xnn! coefficient of xn=1⋅1n!+11!⋅1(n−1)!+12!⋅1(n−2)!+⋯+1n!1=1n!n!n!+n!(n−1)!1!+n!(n−2)!2!+⋯+n!n!=1n! nC0+nC1+nC2+⋯+nCn=2nn!

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

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$\frac{2^{n}}{(n + 1)!}$

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

$n!$

$\frac{2^{n} - 1}{(n - 1)!}$

answer is A.

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Detailed Solution

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

$= (1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \dots + \frac{x^{n}}{n!}) (1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \dots + \frac{x^{n - 1}}{(n - 1)!} + \frac{x^{n}}{n!})$

$coefficient of x^{n}$

$\begin{array}{l} = 1 \cdot \frac{1}{n!} + \frac{1}{1!} \cdot \frac{1}{(n - 1)!} + \frac{1}{2!} \cdot \frac{1}{(n - 2)!} + \dots + \frac{1}{n! 1} \\ = \frac{1}{n!} (\frac{n!}{n!} + \frac{n!}{(n - 1)! 1!} + \frac{n!}{(n - 2)! 2!} + \dots + \frac{n!}{n!}) \\ = \frac{1}{n!} (^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + \dots +^{n} C_{n}) \\ = \frac{2^{n}}{n!} \end{array}$