§ 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.