Problem:
Nine tiles are numbered , respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is , where and are relatively prime positive integers. Find .
Solution:
Each player must select an odd number of odd-numbered tiles. Because there are five odd-numbered tiles available, one player must select three of them, and the other two players must select one each. The probability that the first player selects three odd-numbered tiles is ,for there are ways to select three tiles from the nine available, and there are ways to select three odd-numbered tiles from the five available. Given that this event has occurred, the probability that the second player will choose exactly one odd-numbered tile is , for there are ways to select three tiles from the six that remain, and there are ways to select one odd-numbered and two even-numbered tiles. Given that the first two players have each selected an odd number of odd-numbered tiles, the third is sure to do the same. Because any player can be the one who selects three odd-numbered tiles, the desired probability is , so .
The problems on this page are the property of the MAA's American Mathematics Competitions