2002 Gauss 7 Problem 21

Five people are in a room for a meeting. When the meeting ends, each person shakes hands with each of the other people in the room exactly once. The total number of handshakes that occurs is

\textbf{(A)}\ 5\quad \textbf{(B)}\ 10\quad \textbf{(C)}\ 12\quad \textbf{(D)}\ 15\quad \textbf{(E)}\ 25

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.