Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Prove that any set of $10$ positive integers less than or equal to $100$ will always contain two subsets with the same sum.

Can anyone help me with this problem? Thanks.

share|cite|improve this question
up vote 10 down vote accepted

Let $S$ be a set consisting of ten distinct positive integers, each of them less than or equal to $100$.

How many subsets does $S$ have?

How big can the sum of the elements of $T$ possibly get, for any subset $T\subseteq S$?

By showing that $S$ has more subsets than possible sums-of-subsets, the pigeonhole principle then tells you that there are two distinct subsets whose sums are equal.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.