jmc

Question

Five people are sitting at a round table. Let f 0 be the number of people sitting next to at least 1 female and m0 be the number of people sitting next to at least one male. The number of possible ordered pairs (f,m) is (A) 7 (B) 8 (C) 9 (D) 10 (E) 11

Accepted Answer

Suppose there are more men than women; then there are between zero and two women. If there are no women, the pair is (0,5). If there is one woman, the pair is (2,5). If there are two women, there are two arrangements: one in which they are together, and one in which they are apart, giving the pairs (4,5) and (3,5). All four pairs are asymmetrical; therefore by symmetry there are 8 pairs altogether.