Give a combinatorial proof of the identity n k middot Cn k

Give a combinatorial proof of the identity (n - k) middot C(n, k) = n middot C(n - 1, k)

Solution

Suppose we want to choose a team of k+1 players out of n players and from k+1 one will be the captain.

We can either choose Captain first in n ways (one will be the Captain) and then choose k players from n-1

C(n,1) * C(n-1,k)

or choose k players first and from remaining n-k choose one captain

C(n,k) * C((n-k),1)

They are definitely equal