Are there at least two people present who shook hands with exactly the same number of people? The following is an interview question. > You are invited to a welcome party with $25$ fellow team members. Each of the fellow members shakes hands with you to welcome you. Since a number of people in the room haven't met each other, there is a lot of random handshaking among others as well. If you don't know the total number of handshakes, can you say with certainty that there are at least two people present who shook hands with exactly the same number of people? I have a vague feeling that the answer is yes, which can be justified using the Pigeonhole principle. However, I do not know how to answer the question concre

Proof by contradiction:

Assume that no one shook the same number of hands. That must mean that every possible number of hands shook from $0$ to $24$ occurs for exactly one person.

However, this is impossible, as one person shook $24$ hands, meaning he shook hands with everyone. But there is also someone who shook $0$ hands, shaking no one's hand.