# Handshakes with different people

Jerry is shaking hands with a group of people. There are $$n \geq 2$$ people there. Any person can shake any number of the other peoples' hands, including zero, but any two people can only shake each other's hands once. Shaking hands is a mutual event between exactly two people.

Prove or disprove the claim that there must always be at least two people who shook the same number of hands as each other.

I have two theories for this problem: this is a problem that involves the pigeonhole principle, with there being n-1 holes. Or that it is a problem that involves derangements. I am mainly not sure how to approach the problem.

Go with the pigeon hole proinciple. Each of the $$n$$ people will have shaken between $$0$$ and $$n-1$$ other peoples hands. If someone has shaken $$n-1$$ hands then they have shaken everyone elses hand & so there cannot be someone who has shaken $$0$$ hands.