§ take at most 4 letters from 15 letters.

Trivial: use (150)+(151)+(153)+(154)\binom{15}{0} + \binom{15}{1} + \binom{15}{3} + \binom{15}{4}. Combinatorially, we know that (nr)+(nr1)=(n+1r)\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}. We can apply the same here, to get (150)+(151)=(161)\binom{15}{0} + \binom{15}{1} = \binom{16}{1}. But what does this mean , combinatorially? We are adding a dummy letter, say d1d_1, which if chosen is ignored. This lets us model taking at most 4 letters by adding 4 dummy letters d1,d2,d3,d4d_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.