When do we use multiplication instead of addition? > In a class of $n$ students, how many ways can we choose a size $k$ committee that contains a size m subcommittee? The committee can be chosen in $\binom nk$ ways and subcommittee can be chosen in $\binom km$ ways. So the answer is $\binom nk$$\binom km$. Why do we use multiplication here instead of, say, addition?

Because for each committee of size $k$ there are $\binom {k}{m}$ committees of size $m$. So you need to add $\binom {k}{m}$ as many as the number of committees with size $k$. You have:

$$ \binom {k}{m} + \binom {k}{m} + \dots + \binom {k}{m} $$ This addition has $\binom {n}{k}$ terms, it sums up to $\binom {n}{k}\binom {k}{m}$