Prove the following equality

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

Any help would greatly appreciated.

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- Feb 19th 2008, 06:19 AMhockey777Proof of combination
Prove the following equality

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

Any help would greatly appreciated. - Feb 19th 2008, 07:46 AMtopsquark
- Feb 19th 2008, 08:18 AMPlato
In this case the meaning is $\displaystyle { - r \choose k} = \frac{{\left( { - r} \right)\left( { - r - 1} \right)\left( { - r - 2} \right) \cdots \left( { - r - k + 1} \right)}}{{k!}}$.

The proof is tedious. Expand both sides and compare. - Feb 19th 2008, 09:15 AMhockey777