Question

Twenty-five people go to daily yoga classes at the same gym, which offers eight classes every day. Each attendee wears either a blue, red, or green shirt to class. Show that on a given day, there is at least one class in which two people are wearing the same color shirt.

Answer 1

Therefore, there is at least one class in which two people are wearing the same color shirt

Explanation:

Proof: (By contradiction).

Suppose that, in each of the 8 classes, no two people wear the same color shirt.

Notice that, if there are 4 people in a class, then at least two of them are wearing the same color shirt

because they can only pick between 3 colors.

Hence the number of people in a class must be at most 3.

Since there are 8 classes, then the total number of attendees is at most 24.

A contradiction. Since the problem says there are 25 people attending yoga classes.

Therefore there is at least one class in which two people are wearing the same color shirt