Question: Eleven numbers are chosen from 1, 2, 3, ..., 99, 100. Show that there are two nonempty disjoint subsets of these eleven numbers whose elements have the same sum.
Does anyone know how to solve this question using the pigeonhole principle? I've tried to do so but encountered difficulty in determining which pigeonholes to select.
Hint:
Your set of 11 chosen numbers has $2^{11} - 2 = 2048$ distinct non-empty subsets (pigeons).
The smallest possible sum of a subset is $1$ and the largest is $90 + 91 + \ldots + 99 + 100 = 1045$ (pigeonholes).
Do you see how you can then apply the pigeonhole principle and @Gerry's hint?