Sum of Binomial Coefficients?

Question

Question: Trying to find a proof of the following equation.

For any $m,n\in \mathbb{N}^0$, $$\sum_{k=0}^{m} \binom{n+k}{k} = \binom{n+m+1}{m}.$$

I know that Vandermonde's identity might be useful but not sure where to start.

Answer 1

Hint:

Write it as $$\sum_{k=0}^m \binom {n+k}{n}$$

And then use hockey stick identity

For reference see Hockey stick identity

Using hockey stick identity we get $$\sum_{k=0}^m \binom {n+k}{n}=\binom {n+m+1}{n+1}=\binom {n+m+1}{m}$$