Why, conceptually, is it that $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$? Why, conceptually, is it that $$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}?$$ I know how to prove that this is true, but I don't understand conceptually why it makes sense.

Consider picking $r$ objects out of a set of $n$ of them. Pick a particular object $O$. Either your set has $O$, or it doesn't. There are $\binom{n - 1}{r - 1}$ sets with $O$, and $\binom{n - 1}{r}$ sets without it. (Do you see why those are true?) So together, there are $\binom{n - 1}{r - 1} + \binom{n - 1}{r}$ sets of size $r$.
