Skip to main content

2014 AMC 10B Problem 16

Problem 16 of 25IntermediateCounting & Probability

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

Answer choices

Show solution

Solution

There are 646^4 equally likely ordered outcomes. If exactly three dice show the same value, choose the repeated value in 66 ways, the different value in 55 ways, and the position of the different die in 44 ways. This gives 654=1206\cdot5\cdot4=120 outcomes. If all four dice match, there are 66 outcomes. The probability is 120+664=1261296=772\frac{120+6}{6^4}=\frac{126}{1296}=\frac7{72}. Thus, the correct answer is B .

More practice

Concepts: dice (probability) · casework

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.