06/18/2020 (2011: National Target Round, Problem 7)

Q: The four faces of a fair tetrahedral die are numbered one, two, three and four. Each time the die is rolled, three numbers are visible. If the die is rolled three times, what is the probability that the sum of all nine of the numbers showing is 21? Express your answer as a common fraction.

A: 5/32

The ways you can get 21 from 3 sets of 3 numbers is:

{2, 3, 4}, {1, 2, 3}, {1, 2, 3}

{1, 2, 3}, {1, 2, 3}, {2, 3, 4}

{1, 2, 3}, {2, 3, 4}, {1, 2, 3}

{1, 2, 4}, {1, 2, 4}, {1, 2, 4}

{1, 3, 4}, {1, 2, 4}, {1, 2, 3}

{1, 3, 4}, {1, 2, 3}, {1, 2, 4}

{1, 2, 4}, {1, 3, 4}, {1, 2, 3}

{1, 2, 4}, {1, 2, 3}, {1, 3, 4}

{1, 2, 3}, {1, 2, 4}, {1, 3, 4}

{1, 2, 3}, {1, 3, 4}, {1, 2, 4}

There are 10 ways to roll a 21 out of the total 4³ = 64 total combinations. 10/64 = 5/32.