22 people attend a party. Each person shakes hands with at most 20 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?
If each person shakes hands with exactly 20 other people, then there will be $\frac{22 \cdot 20}{2} = \boxed{220}$ handshakes, since two people are required to complete a handshake. To achieve 220 handshakes, we arrange the participants in a circle. Each person shakes hands with everyone except the person directly across from himself or herself (and himself or herself).