Question:
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?

Answer:
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).