Identity proof with binomial coefficients

Question

I have to prove the following identity and have to idea how to start.

$ { {-n} \choose k} = (-1)^k {{n+k-1} \choose k} $

I know that $ \sum_{k=0}^{n} (-1)^k{n \choose k}=0 $

but I cannot see a way, in which it could help prove the mentioned above.

Thanks for your help

Answer 1

We can formally write $\binom {-n}{k}$ with factorial formula: $$ \binom {-n}{k}=\frac {(-n)(-n-1)(-n-2)\dotsm (-n-k+1)}{k!} =\\ (-1)^{k}\frac {(k+n-1)(k+n-2)\dotsm (n)}{k!}=(-1)^{k}\binom {k+n-1}{k} $$