How to prove $\sum_{1 < k < n}\binom{n-k}{1}\binom{k-1}{t+1} = \binom{n}{t+3}$?

Question

How to prove $\sum_{1 < k < n}\binom{n-k}{1}\binom{k-1}{t+1} = \binom{n}{t+3}$?

I encountered this in a book and don't know how they did this. Here $t$ is a constant.

Answer 1

In order to choose $t+3$ items from $\{1,\ldots,n\}$, you can first pick any item $k$ as the second-largest (where $1\le k<n$), then choose the largest from the upper $n-k$ elements, and then choose the remaining $t+1$ items from the lower $k-1$ numbers.