5 people enter a racquetball tournament in which each person must play evert other person exactly once. Determine the total number of games that will be played

This is the "handshake problem", namely with n people, how many handshakes will there be if every will shake hands with everyone else.

n people will shake hands with (n-1) other people. Since we are counting twice for each handshake, the number of handshakes is n (n-1) / 2.

For n=5, the number of matches is 5 (5-1) / 2=10.

This is also the number of diagonals in an n-sided convex polygon.