Party size from 120 handshakes

At a party every pair of guests shakes hands exactly once, and $120$ handshakes take place in total. How many guests are there?

16. The number of handshakes is the number of pairs, $\binom{N}{2}=\tfrac{N(N-1)}{2}$. Setting this to $120$ gives $N(N-1)=240=16\cdot15$, so $N=16$.