3 dice are rolled. What is the probability that you will get the sum of the…

For independent trials like rolling several dice, probability equals the number of favorable outcomes divided by the total number of equally likely outcomes. To count favorable outcomes for a target sum, list every distinct combination of face values that adds to that sum, then multiply each combination by the number of ways it can be arranged among the three dice: a combination of three different digits has 3! = 6 arrangements, while a combination with one digit repeated has only 3!/2! = 3 arrangements.

1. Total possible outcomes when three dice are rolled: 6 × 6 × 6 = 63 = 216.

2. List every combination of values from 1 to 6 that adds up to 10: (1,3,6), (1,4,5), (2,3,5), (2,4,4), (3,3,4), (2,2,6).

3. The combinations (1,3,6), (1,4,5) and (2,3,5) each have three different digits, so each one can be arranged in 3! = 6 ways: 6 + 6 + 6 = 18 outcomes.

4. The combinations (2,4,4), (3,3,4) and (2,2,6) each have one digit repeated, so each one can be arranged in only 3!/2! = 3 ways: 3 + 3 + 3 = 9 outcomes.

5. Total favorable outcomes = 18 + 9 = 27, so the required probability is 27/216.

As a check, the possible sums with three dice range from 3 to 18 with a mean of 10.5, so sums 10 and 11 are symmetric around the mean and must have an equal count of favorable outcomes. Working out the sum-11 case the same way also gives 27 favorable outcomes, confirming 27/216 is correct here.