Question

Two cubical dice each have removable numbers $1$ through $6.$ 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 $7?$

$\dfrac{1}{9}$

$\dfrac{1}{8}$

$\dfrac{1}{6}$

$\dfrac{2}{11}$

$\dfrac{1}{5}$

Solution:

Randomly attaching the tiles and then rolling is equivalent to choosing two of the twelve numbers at random and adding them.

Suppose the first top face shows $N.$ For a sum of $7,$ the second must be $7 - N,$ and there are exactly $2$ tiles equal to $7 - N$ among the remaining $11.$

So the probability is $\dfrac{2}{11}.$

Thus, the correct answer is D.

2009 AMC 10A Problem 22

Solution:

Problem 22 in Other Years