combinatorics - Show $sum_{k=0}^{m} binom{n-k}{m-k}=binom{n+1}{m}$

My question is:

show $$\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$$
$$n\geq m\geq 1$$
I tried to do this via induction and failed. there has to be another way of doing this.

We could either explain combinatorically that we are counting the same thing in 2 different ways, in this sense, we are counting the number of ways to choose $m$ elements from a set with $n+1$ elements.

Or we could try like me to prove it algebraically, both ways are valid...I'd like some help, or a push in the right direction.

Answer

Hint: The right side counts the number of ways to pick $m$ elements out of $n+1$.

For the left: partition the $m$-element subsets of $n+1$ based on the highest element that is NOT included in the subset.