Question

show that CoCr+C1Cr+1+C2Cr+2+...........+Cn-rCn answer s explanation​

Answer 1

Answer:2nCn + r = {2n}!/{n + r}!{n - r}!

Step-by-step explanation:

(1 +  x)n = C0 + C1x + C2x2 + ....+ Crxr + Cr + 1xr + 1 + .... + Cnxn

and (1 + 1/x)n = C0 + C1/x + C2/x2 + .... + Cr/xr + Cr + 1/xr + 1 + ... + Cn/xn

In the product of these two expansions, collecting the coefficient of xr

C0Cr + C1Cr + 1 + C2Cr + 2 + ...+ Cn - rCn

= coefficient of xr in (1 + x)2n/xn

= coefficient of xn + 1 in (1 + x)2n

=  2nCn + r = (2n)!/(n + r)!(n - r)!