Found a pattern on Pascal's triangle: ${n \choose r}=\sum_{i=0}^k {k \choose i}{n-k \choose r-k+i}$. Has anyone come across it anywhere?

Question

In a standard Pascal's triangle, the entry in the $n$th row and $r$th column denoted as ${n \choose r}$ can be expressed using $k+1$ entries $k$ rows above itself as $${n \choose r}=\sum_{i=0}^k {k \choose i}{n-k \choose r-k+i}$$ where $0\le k\le r$.

A simple example to help visualize the identity:

Feel free to play around with values on Desmos calculator