Three dice are rolled simultaneously. What is the probability that exactly two of the dice will come up as the same number?

A. 5/12
B. 11/24
C. 25/54
D. 13/27
E. 1/2
_________________

There are a total of 6 x 6 x 6 = 216 outcomes.

Let’s say the two numbers that come up are 1 and 2, and then we could have any of the following:

(1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 2, 2), (2, 1, 2) and (2, 2, 1)

However, we have 6C2 = (6 x 5)/2 = 15 ways to choose two distinct numbers from six. For each of these 15 pairs of numbers, we have 6 ways (see the example for 1 and 2 above). Therefore, we have a total of 15 x 6 = 90 ways that exactly two of the dice will come up as the same number. So the probability is 90/216 = 15/36 = 5/12.

Alternate Solution:

Suppose the first die is rolled. The second die can either match the first die (a probability of 1/6) or it can differ from the first die (a probability of 5/6).
If the second die matches the first die, the only way we can meet the condition of having “exactly two of the dice come up as the same number” is if the last die differs from the first two dice, which has a probability of 5/6. The probability of meeting the condition under this scenario is 1/6 x 5/6 = 5/36.

Alternatively, if the second die differs from the first die, then we can meet the required condition by having the third die match any one of the previous two dice; which is a probability of 2/6 = 1/3. The probability of meeting the condition under this scenario is 5/6 x 1/3 = 5/18.

In total, the probability of meeting the condition is 5/36 + 5/18 = 15/36 = 5/12.

Answer: A
_________________