During a party, a total of 78 handshakes occurred. If each person shook hands once with each of the other people, how many people were at the party?
Since each person shakes hands with each other person, every pair of people will shake hands once. So, 78 represents the number of pairs, which we can count as ${n \choose 2}$ where $n$ is the number of people at the party. So, $n(n-1) = 2 \cdot 78 = 2 \cdot 6 \cdot 13 = 12 \cdot 13$. So, $n=13$ gives us $\boxed{13}$ people at the party.