Problem:
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is , where and are relatively prime positive integers. Find .
Solution:
Answer (191):
The required probability is the same as the probability that the five men get paired with women when the people are randomly paired with each other. The number of ways of pairing people is . The number of ways of pairing each man to a woman and then pairing the remaining four women to each other is . Because each pairing is equally likely, the required probability is
The requested sum is .
The number of ways for the five men to choose positions in which to stand is . For men not to stand diametrically opposite each other, the first man can stand in any of positions, the second man can then stand in any of positions, the third in any of positions, the fourth in any of positions, and the fifth in any of positions. The probability is therefore
as above.
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions