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Q.

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

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a

2nn!

b

2n−1(n−1)!

c

n!

d

2nn+1!

answer is A.

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Detailed Solution

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 xn in 1+x1!+x22!+⋯+xnn!2