## § take at most 4 letters from 15 letters.

Trivial: use $\binom{15}{0} + \binom{15}{1} + \binom{15}{3} + \binom{15}{4}$.
Combinatorially, we know that $\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$.
We can apply the same here, to get $\binom{15}{0} + \binom{15}{1} = \binom{16}{1}$.
But what does this *mean*, combinatorially? We are adding a dummy letter, say $d_1$,
which if chosen is ignored. This lets us model taking at most 4 letters by adding
4 dummy letters $d_1, d_2, d_3, d_4$ and then ignoring these if we pick them up; we
pick 4 letters from 15 + 4 dummy = 19 letters.
I find it nice how I used to never look for the combinatorial meaning behind
massaging the algebra, but I do now.