HMMT February

S is the set of positive integers with exactly three ones in its binary representation. The number of such integers with at most $d$ total bits is $\binom{d}{3}$$, and noting that \binom{9}{3}=84$ and $\binom{10}{3}=120$, we want the 16th smallest integer of the form 2^{9}+2^{x}+2^{y}$, where$y<x<9$. Ignoring the$2^{9}$term, there are \binom{d^{\prime}}{2}$ positive integers of the form $2^{x}+2^{y}$ with at most $d^{\prime}$ total bits. Because $\binom{6}{2}=15$$, our answer is$2^{9}+2^{6}+2^{0}=577$. (By a bit, we mean a digit in base 2 .)