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.