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Math Help - Proving Combinatorics Summation Identitiy

  1. #1
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    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.
    Last edited by Plato; September 14th 2011 at 02:53 AM. Reason: LaTeX fix
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  2. #2
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    Re: Proving Combinatorics Summation Identitiy

    Quote Originally Posted by RooneyOwns12 View Post
    \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}
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