I'm not sure what you would consider combinatorics, and what you wouldn't. The standard solution mentioned in other comments seems combinatorial to me. The insight necessary is that the pills are not "atomic", they can be divided.
One non-standard way to solve, which is kind of the same, is to smash the pills into infinitesimally small bits and then blend them until the particle distribution is uniform and well mixed, then eat half of the pile. with probability 1, you win.
The fact that doing this subdivision and recombination in the most coarse possible way ( break in two and recombine any way you like, then eat half ) actually works with probability 1 as well seems like a numerical miracle, which is kind of what I like to think combinatorics is.