Every face exactly six times in 36 rolls

Question

A fair die is rolled 36 times. What is the probability that each of the six faces comes up exactly six times?

Short answer

$\dfrac{36!}{(6!)^6\,6^{36}}\approx 2.6\times10^{-4}$ . This is a multinomial probability: the number of sequences with six of each face, times the probability $6^{-36}$ of any one sequence.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Counting and Combinatorics
Expectation, Mean, and Variance

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