Why does ${n \choose k }{k \choose l} = {n \choose l}{n - l \choose k - l}$

Question

Why does (in words) $${n \choose k }{k \choose l} = {n \choose l}{n - l \choose k - l}$$

Using the binomial theorem, it's easy to show equality. I just don't understand how to equate the two in words. The LHS appears to be "The number of subsets of size $k$ from a set of size $n$ times the number of subsets of size $l$ from a set of size $k$. It is not obvious to me how the right side is counting the same thing.

Any hints are helpful, thanks

Answer 1

A club has $n$ members. They have to choose a committee of $k$ members, and a sub-committee of $l$ members.

• Method 1: choose the committee, $\binom nk$ ways, then the subcommittee, $\binom kl$ ways.
• Method 2: choose the subcommittee first, $\binom nl$ ways, then another $k-l$ committee members from the remaining $n-l$ club members, $\binom{n-l}{k-l}$ ways.

Therefore $$\binom nk \binom kl = \binom nl \binom{n-l}{k-l}\ .$$