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Question

The coefficient of x5 in the expansion of (1+x)21+(1+x)22+..+(1+x)30 is

A
51C5
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B
9C5
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C
31C621C6
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D
30C5+20C5
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Solution

The correct option is D 31C621C6
Let
S=(1+x)21+(1+x)22+(1+x)23+...(1+x)30 ...(i)

(1+x)S=(1+x)22+(1+x)23...(1+x)30+(1+x)31 ...(ii)

Subtracting (i) from (ii), we get

xS=(1+x)31(1+x)21

S=(1+x)31x(1+x)21x

Coefficient of xr

=31Crxr121Crxr1

=(31Cr21Cr)xr1 ... (a)

Therefore the coefficient of x5 implies

r1=5

r=6

Substituting in (a), we get coefficient of x5

=(31C621C6)

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