Where Error in evaluate summmation :$$\sum_{k=0}^{m}\binom{n-k}{m-k}$$
I know that : $$\sum_{k=0}^{m}\binom{n-k}{m-k}=\binom{n+1}{m}$$
But I does't know error My proof this : we have : $$\binom{n-k}{m-k}=\binom{n+1-k}{m-k}-\binom{n-k}{m-k-1}$$
Let :$$a_k=\binom{n+1-k}{m-k}$$
Therfore: $$\binom{n-k}{m-k}=a_k-a_{k+1}$$ This implies that : $$\sum_{k=0}^{m}\binom{n-k}{m-k}=a_0-a_{m+1}=\binom{n+1}{m}-\binom{n-m}{-1}$$ I think this is wrong, because I do not know the exact value of this :$\binom{n-m}{-1}$I appreciate your interest
If anyone has another interesting method I'd be happy to share it
