06/07/2020 (2011: National Sprint Round, Problem 19)

Q: What is the probability that a list of five different, single-digit numbers, whose sum is 31, will contain exactly two prime numbers? Express your answer as a common fraction.

A: 3/5

We list out the possible lists of 5 different single-digit numbers that add up to 31and we get:

{4, 5, 6, 7, 9}

{3, 5, 6, 8, 9}

{2, 5, 7, 8, 9}

{1, 6, 7, 8, 9}

{3, 4, 7, 8, 9}

Out of these 5 lists, 3 of them ({4, 5, 6, 7, 9}, {3, 5, 6, 8, 9}, {3, 4, 7, 8, 9}) have exactly two prime numbers. That means the answer is 3/5.