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The coefficient of $x^{n}$ $i n t h e e x p a n s i o n o f$ ${(1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} \dots + \frac{1}{n!} x^{n})}^{2}$

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a

2nn!

b

2nn

c

n!

d

1n!

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detailed solution

Correct option is A

Coefficient of xn in1+11!x+12!x2+…+1n!xn2= coefficient of xn is1+11!x+12!x2+…+1n!xn+1(n+1)!xn+1…2= coefficient of xn in = e2x=1+2x+22x22!+23x33!⋯=2nn!Alternate Solution Coefficient of xn in1+11!x+12!x2+⋯+1n!xn 1+11!x+12!x2+⋯+1n!xn =1n!+11!(n−1)!+12!(n−2)!+⋯+ =1n! nC0+nC1+nC2+⋯nCn−1+nCn=2nn!

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