Problem:
A hotel packed a breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls, and, once they were wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability that each guest got one roll of each type is , where and are relatively prime positive integers, find .
Solution:
The probability that the first bag contains one of each of the three types of rolls is . The probability that the second bag will then contain one of each is . If the first two bags have a complete selection, then the last bag must too. Thus the probability that all three breakfasts have a complete selection is , and .
The problems on this page are the property of the MAA's American Mathematics Competitions