Binomial Q (1 Viewer)

[GaganDeep]

Show that (nCr)nCn-1)=(n-r+1)
Hence find the sum of
nC1:nC0 + 2(nC2):nC1 +3(nC3):nC2 + ...+n (nCn):nCn-1
I can do first part but can't do the sum

 

[Riviet]

I think the first part was supposed to read:
Show that r(nCr)/{nC(r-1)} = n-r+1

Sum = (n-1+1) + (n-2+1) + ... + (n-(n-1)+1) + (n-n+1) [using first part]
= n + (n-1) + (n-2) + ... + 2 + 1

This is a sum of an AP with n terms, d=-1, a=n

.'. Sum = n/2.(2a + (n-1)d)
=(n/2).(2n + (n-1)(-1))
=(n/2).(2n-n+1)
=(n/2).(n+1)