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C0CR +C1CR+1 +C2CR+2....+CN-RCN= 2N!/(N-R)!(N+R)!
PROVE..
this can be written as C0Cn-r + C1Cn-r-1 + C2Cn-r-2 +..........+ Cn-rC0
to get this sum you can consider (1+x)n(x+1)n
when you expand the two and multiply the terms you will find that the coefficient of xn-r is the required sum.
Hence to get the sum you only need to find the coefficient of xn-r in (1+x)n(x+1)n i.e. (1+x)2n i.e 2nCn-r
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