If $n$ people are placed in a room, then at least $2$ of those people will have the same number of friends in the room.
I want to prove this statement.
Here are some of my thoughts:
If all the people are strangers, then clearly everyone has zero friends in the room. Hence they all have the same number of friends. Thus, clearly there exists at least one pair in the room with the same number of friends. I see this as the trivial case.
Maybe I can prove this by assuming that, for any collection of $n$ people, all the people have a different number of friends; perhaps I can then find a contradiction.
Proceeding with the idea in 2., I devised the following relationship:
person $1$ has $0$ friends, person $2$ has $1$ friend, person $3$ has $2$ friends, ... , person $(n-1)$ has $(n-2)$ friends, person $n$ has $(n-1)$ friends
Now I want to find a way to use this relationship to show that a contradiction arises, i.e. at least $2$ people must have the same number of friends in the room.