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Question

The value of the expression C20C21+C22+(1)nC2n

A
0 if n is odd
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B
(1)n if n is odd
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C
(1)n2 nCn2 if n is even
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D
(1)n1 nCn1 if n is even
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Solution

The correct option is C (1)n2 nCn2 if n is even
When n is odd, taken n = 2m + 1, so that
S=C20C21+C22+(1)2mC22m+(1)2m+1C22m+1
=(C20C22m+1)(C21C22m)+
But 2m+1C0=2m+1C2m+1,
2m+1C1= 2m+1C2m etc.
Therefore S = 0
When n is even, we take n = 2m. In this case
C20C21+C22C23++(1)2mC22m
= Coefficient of constant term in
[C0C1x+C2x2+(1)2mC2mx2m]
[C0+C11x+c21x2++C2m1x2m]
= Coefficient of constant term in
(1x)2m(1+1x)2m
= Coefficient of x2m in (1x)2m(1+x)2m
= Coefficient of x2m in (1x2)2m
=(1)m(2mCm)=(1)n2(nCn2)

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