Suppose $45$ handshakes occurred in a room, how many people were in the room?
Someone asked me this question and I was going to answer him using graph theory and my knowledge of the number of vertices in a completed graph. But then he told me this is for a statistics class and I can't use any of those. I don't work with discrete math a lot, so the answer did not come to me immediately. Can someone show me a fundamental way or perhaps clever way to solve this by not brutally counting?
Assume there are $n$ people in the room, and no pair shook hands more than once. If everybody shook hands with everybody else, there would be ${n \choose 2} = n(n-1)/2$ handshakes. This is $45$ for $n = 10$, and less than $45$ if $n < 10$. So there are at least $10$ people in the room.