TITLE: combinatorics: The pigeonhole principle
QUESTION [5 upvotes]: Assume that in every group of 9 people, there are 3 in the same height.
Prove that in a group of 25 people there are 7 in the same height.
I started by defining:
pigeonhole- heights.
pigeons-people.
I do not know how to use the assumption.
thanx.

REPLY [2 votes]: Just for fun an alternative to Jyrki's nice solution:
If there are at most four different heights, then $25 = 6\cdot 4 + 1$ shows that seven people have the same height.
So let's assume that there are five people all of different heights.
If among the other 20 people four have different heights, the group of these four plus the five give a group of nine contradicting the hypothesis as at most two in this group of nine have the same height.
So the remaining 20 people have at most three different height. Now $20 = 6\cdot 3 + 2$ shows that seven people of the remaining 20 have the same height.