Anyone know how to do this?
In real life, if a person A is a friend of a person B then B is a friend of A. Let now S be the set of students in our department. Prove that there are at least two students in S having the same number of friends among the same set of people S
Suppose that there are in total of $n=|S|$ people in the department. For an arbitrary person, the number of friends he can have belong to the set $A=\{1,2,3,...,n-2,n-1\}$ (assume that every person can befriend with himself). Since there are $n$ people while $|A|=n-1$, at least 2 people have the same number of friends.
If we assume that no one can befriend with him/herself, then $A=\{0,1,2,...,n-1\}$. If we suppose that there's at least 1 person who has no friend, then $A'=\{0,1,2,...,n-2\}$ since there cannot be any people with $n-1$ friend now, making $|A'|=n-1$ anyway.
Assuming one can't befriend himself:
Let $n=|S|$. Let's suppose there's a situation where all the students have a different number of friends. Then since any student can befriend at most $n-1$ people, there is exactly one student having $i$ friends for $0 \leq i <n$. That means that there is a student who befriended everyone else. In particular, he befriended the student with no friends, which is a contradiction, since if A is a friend of B, B is a friend of A.
