Intuitive explanation of "reverse" binomial coefficient formula ${n\choose k}=\sum_{r=0}^s {n-s \choose k-r}{s\choose r}$

Question

We have the formula $$ {n\choose k} = {n-1\choose k-1}+{n-1\choose k} $$ and it is not at all difficult to prove by induction that $$ {n\choose k}=\sum_{r=0}^s {n-s \choose k-r}{s\choose r}. $$ My question is: how to understand this formula intuitively, presumably with some situation in combinactorics?

Answer 1

How can we choose $k$ elements out of $n$?

Well, fix $s$. We want to choose some $(=k-r)$ elements out of the first $n-s$ of them, and them some more $(=r)$ out of the next $s$ of them.
This approach gives us $ \sum {n-s \choose k-r} { s \choose r}$.