TITLE: Prove that 2 students live exactly five houses apart if
QUESTION [11 upvotes]: There are 50 houses along one side of a street. A survey shows that 26 of these houses have students living in them. Prove that there are two students who live EXACTLY five houses apart on the street.
How do I use the pigeonhole principle for this question?

REPLY [6 votes]: You can also divide the houses in 25 groups of 2 houses:
$(1,6), (2,7), (3,8), (4,9), (5,10)$ and add 10, 20 , 30, 40 for the other $4*5=20$ groups.
26 students, 25 pairs....