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Question

The coefficient of x50 in the binomial expansion of (1+x)1000+x(1+x)999+x2(1+x)998+...+x1000 is :

A
(1000)!(49)!(951)!
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B
(1001)!(51)!(950)!
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C
(1000)!(50)!(950)!
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D
(1001)!(50)!(951)!
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Solution

The correct option is D (1001)!(50)!(951)!
S=(1+x)1000+x(1+x)999+x2(1+x)998+...+x1000

This is a Geometric Series.

a=(1+x)1000

r=xx+1

n=1001

S=((1+x)1000)(1x1001(x+1)1001)1x1+x=(1+x)1001x1001

Coefficient of x50 is 1001C50=(1001)!(951)!(50)! .

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