# Math Help - Proving Combinatorics Summation Identitiy

1. ## Proving Combinatorics Summation Identitiy

$\sum_{k = i}^{n}\binom{n}{k}\binom{k}{i}(-1)^{n-k} = 0 \text{ for } i < n$

I tried some cases, like with n= 4, k = 3, and i = 1, and i got like -4 + 12 -12 +4 = 0 (seems symeetrical), but I am not sure where to go to actually prove this identity.

2. ## Re: Proving Combinatorics Summation Identitiy

Originally Posted by RooneyOwns12
$\sum_{k = i}^{n}\binom{n}{k}\binom{k}{i}(-1)^{n-k} = 0 \text{ for } i < n$

I tried some cases, like with n= 4, k = 3, and i = 1, and i got like -4 + 12 -12 +4 = 0 (seems symeetrical), but I am not sure where to go to actually prove this identity.
Use the fact $\binom{n}{k}\binom{k}{i} = \binom{n}{i}\binom{n-i}{k-i}$