I am trying to prove algebraically and using combinatorics that
$$\sum_{i=0}^r \binom{i + m -1}{m - 1} = \binom{m + r}{m}$$ where $m\geq1$ and $r\geq0$
The algebraic proof went smoothly but I am confused as to what the left hand side means. I understand the right hand side is showing how to choose $m$ elements out of $m+r$ elements, but how does that relate to the left hand side?