Problem:
A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is , where and are relatively prime positive integers. Find .
Solution:
Let be an integer and let the bag contain distinct pairs of tiles. The probability that two of the first three tiles selected make a pair is
Now let be the probability of emptying the bag when the bag initially contains distinct pairs of tiles. Then and for ,
Using this recursion formula repeatedly, we find that
Setting we have
The sum of the numerator and denominator is .
The problems on this page are the property of the MAA's American Mathematics Competitions