1998 AIME Problems/Problem 4
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
There are 5 odd tiles and 4 even tiles. In order for a player to have an odd sum, she must have an odd number of odd tiles. Thus, the only possibility is that one player gets three odd tiles and the other two players get two even and one odd tile. We count the number of ways this can happen. (Below, we say that the people are distinguishable - say that they are named Ana, Beth, and Clark, for example.)
We have 3 choices for the player who will get three odd tiles, and 
choices for the tiles that he gets. The other two odd tiles can be distributed to the other two players in 2 ways, and the even tiles can be distributed between them in 
ways. This gives us a total of 
possibilities in which all three people get odd sums.
In order to calculate the probability, we need to know the total number of possible distributions for the tiles. The first player needs three tiles which we can give him in 
ways, and the second player needs three of the remaining six, which we can give him in 
ways. Finally, the third player will simply take the remaining tiles in 
way. So, there are 
ways total to distribute the tiles.
Thus, the total probability is 
so the answer is 
.
See also
| 1998 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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