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Math Help - Proof of combination

  1. #1
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    Proof of combination

    Prove the following equality

    \binom{-r}{k}= (-1)^{k} \binom{r+k-1}{k}

    Any help would greatly appreciated.
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  2. #2
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    Quote Originally Posted by hockey777 View Post
    Prove the following equality

    \binom{-r}{k}= (-1)^{k} \binom{r+k-1}{k}

    Any help would greatly appreciated.
    How are you defining {-r \choose k}? Normally the combinitorial function is not defined for negative values.

    -Dan
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  3. #3
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    In this case the meaning is { - r \choose k}  = \frac{{\left( { - r} \right)\left( { - r - 1} \right)\left( { - r - 2} \right) \cdots \left( { - r - k + 1} \right)}}{{k!}}.
    The proof is tedious. Expand both sides and compare.
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  4. #4
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    Quote Originally Posted by Plato View Post
    In this case the meaning is { - r \choose k}  = \frac{{\left( { - r} \right)\left( { - r - 1} \right)\left( { - r - 2} \right) \cdots \left( { - r - k + 1} \right)}}{{k!}}.
    The proof is tedious. Expand both sides and compare.
    = -1^k \frac{n(n+1)...(n+k-1)}{k!}

    = -1^k \binom{n+k-1}{k}

    Would that work?
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