I'm looking for an intuitive explanation for this identity: ${n \choose k} = \frac{n}{k}{n-1 \choose k-1}$ for $0 < k \leq n$.
The math adds up, but I can't see why it's true. I'd expect that choosing $k$ elements from an $n$-set would be like choosing $k-1$ elements from an $(n-1)$-set, then add an $n$th element to the set, and choose another element from the $n - k + 1$ not-yet-chosen elements. But I guess I'm missing something.
Can anyone provide some intuition?