PROVE THAT CoCr+ C1Cr+1 + .... + Cn-rCn = (2n)! / (n-r)! (n+r)!

Dear student

(1+x)n=C0n+C1nx+....+Cnnxn(1+1/x)n=C0n+C1n(1/x)+....+Cnn1/xngiven summation= coefficient of xr in (1+x)n(1+1/x)n=coefficient of xr in (1+x)2nxn=coefficient of xn+r in (1+x)2n =Cn+r2n=2n!(n+r)!(2n-n-r)!=2n!(n+r)!(n-r)!

Regards

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