Binomial coefficient identity

Question

Is there an identity $\sum_{r=0}^{l-1-p} \binom {r}{p} = \binom{l}{p+1}$ ? I need a proof for this, if it holds. For $l=2$ I can see that it is true.

Answer 1

I'd rather state: $$ \sum_{r=p}^{m}\binom{r}{p}=\binom{m+1}{p+1},$$ that can be proven by induction on $m$, since: $$\binom{m+1}{p+1}=\binom{m}{p}+\binom{m}{p+1}.$$