Problem:
Two cubical dice each have removable numbers through . The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is ?
Answer Choices:
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Solution:
Suppose that the two dice originally had the numbers and , respectively. The process of randomly picking the numbers, randomly affixing them to the dice, rolling the dice, and adding the top numbers is equivalent to picking two of the twelve numbers at random and adding them. There are sets of two elements taken from . There are ways to use a and to obtain , namely, , and . Similarly there are ways to obtain the sum of using a and , and ways using a and . Hence there are pairs taken from whose sum is . Therefore the requested probability is .
Because the process is equivalent to picking two of the twelve numbers at random and then adding them, suppose we first pick number . Then the second choice must be number . For any value of , there are two "removable numbers" equal to out of the remaining , so the probability of rolling a is .
The problems on this page are the property of the MAA's American Mathematics Competitions