Use the pigeonhole principle to prove that if $n\geq 2$ people go to the same party, there are at least 2 people who shake hands with the same number of other people.
Hint: Take the set of people at the party as your domain, define a function that evaluates how many people each person shook hands with.
Need help with this question, kind of stuck. How do I approach and proof the problem? I know the one-to-one function is a contrapositive and that's about it.
HINT
You have $n$ people, who can at most shake the hands of $n-1$ people. Now, if all $n$ people shake a different number of hands, then you have one person shaking $0$ hands, one person shaking $1$ hand, one person shaking $2$ hands ... and one person shaking $n-1$ hands ... what's wrong with that picture?