TITLE: Sum of binomial coefficients $\sum_{k=0}^{n}\binom{k+m}{m}$
QUESTION [5 upvotes]: Let $n,m \in \mathbb{N}$
Evaluate:
$$\sum_{k=0}^{n}\binom{k+m}{m}$$
I don't know what to do, because none of the identities I know match this sum.
Help is greatly appreciated.

REPLY [5 votes]: Consider the recurrence identity
$$
   \binom{n+1}{m+1} - \binom{n}{m+1} = \binom{n}{m}
$$
for $n=k+m$. We get, by telescoping:
$$ \begin{eqnarray}
   \sum_{k=0}^n \binom{k+m}{m} &=& \sum_{k=0}^n \left( \binom{m+k+1}{m+1} - \binom{m+k}{m+1} \right)  = \sum_{k=0}^n \left(F(k+1)-F(k) \right)  \\ &=&  F(n+1)-F(0) = \binom{m+n+1}{m+1} - \binom{m}{m+1} \\ &=& \binom{m+n+1}{m+1} 
\end{eqnarray}$$