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Question

If (1+x)n=C0+C1x+C2x2+...+Cnxn, then C0C2+C1C3+C2C4+...+Cn2Cn=

A
(2n)!(n!)2
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B
(2n)!(n1)!(n+1)!
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C
(2n)!(n2)!(n+2)!
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D
None of these
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Solution

The correct option is C (2n)!(n2)!(n+2)!
(1+x)n=C0+C1x+C2x2+C3x3+C4x4+...+Cn2xn2+Cn1xn1+Cnxn ...(1)
(x+1)n=C0xn+C1xn1+C2xn2+C3xn3+C4xn4+...+Cn2x2+Cn1x1+Cn ...(2)
Multiplying (1) and (2) and equating the coefficients of xn2, we get
C0C2+C1C3+C2C4+...+Cn2Cn
= the coefficients of xn2 in (1+x)2n
=2nCn2=(2n)!(n2)!(n+2)!

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