C0CR +C1CR+1 +C2CR+2....+CN-RCN= 2N!/(N-R)!(N+R)!

PROVE..

this can be written as C₀C_n-r + C₁C_n-r-1 + C₂C_n-r-2 +..........+ C_n-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 x^n-r is the required sum.

Hence to get the sum you only need to find the coefficient of x^n-r in  (1+x)ⁿ(x+1)ⁿ  i.e. (1+x)²ⁿ i.e ²ⁿC_n-r