I hope you would.

Consider the same situation for a group of 4 people and try to extend the reasoning from the case of 3 and 4 to n.

You are considering functions from {1, ..., |X|} to {0, ..., |X| - 1} that map people to the number of their friends. You can show that |X| - 1 or 0 is not in the range of such function. This makes pigeonhole principle applicable: the set of pigeons is the domain and the set of holes is the range. More formally, pigeonhole principle says that any function with domain A and range B where |A| > |B| cannot be an injection, and this is precisely the situation that you have here.