Show that some 5 consecutive chairs must be occupied.

Question

A group of 25 people are seated in a row of 30 chairs. Show that some 5 consecutive chairs must be occupied.

Answer 1

Since there are 25 people and 30 chairs, there are 5 empty seats. Divide the row of 30 chairs into 6 groups of 5 consecutive seats. Since there are 6 groups and 5 empty chairs, at least one of the 6 groups has no empty seats. So in that group with no empty seats, all five consecutive seats are full.

Answer 2

Place single empty chairs so there's 4 full chairs between them. Obviously they can't be pushed further apart, as a group of 5 full chairs would then exist. How much do you have stuck on the end after you've placed them all?