Problem:
Five people are sitting at a round table. Let be the number of people sitting next to at least one female and be the number of people sitting next to at least one male. The number of possible ordered pairs is
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Solution:
Let denote the number of females in the group of . If then is the only pair, and if then is the only pair. If then the possible pairs are and , depending on whether or not the females are sitting next to each other. By symmetry, the pairs , and and correspond to , and , respectively. Thus there are possible pairs.