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Question

C0C1+C1C2+Cn1C2=(2n)!(n1)!(n=1)!

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Solution

L. H. S. are respectively than coefficient of x,x2 and xr in 1xn(1+x)2n or the coefficient of xn+1,xn+2,xn+r in (1+x)2n. These are 2nCn+1,2nCn+2,2nCn+r respectively

or (2n)!(n1)!(n+1)!,(2n)!(n2)!(n+2)! and (2n)!(nr)!(n+r)!

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