# Pigeon Hole Principle

• April 8th 2010, 12:08 PM
Pigeon Hole Principle
Alright, here is my problem:

30 buses are to be used to transport 2000 refugees from Location A to Location B. Each bus has 80 seats. Assume one seat per passenger. Prove that one of the buses will have at least 14 empty seats.

Now.. $\frac{2000}{80}=25$ which means we wont need to use the 30 buses or.. we will have a certain amount of people in each bus to use all of the buses. But.. where do I go from here? Any help would be appreciated!
• April 8th 2010, 12:28 PM
MoeBlee
80*30 = 2400 seats

2400-2000 = 400 empty seats

400/30 = 13.3... empty seats per bus
• April 11th 2010, 03:27 AM
baz
Minimum seats in each bus will be left only if you were to take at least 2000/30 = 66.667 persons per bus, which means 80-66.667=13.33 empty seats per bus.
So at least one of the bus must have 14 empty seats beacuse maximum number of buses which can take 67 passengers per bus are 29 (67x2=1943). Which means 30 bus has 23 empty seats.