Three dice in strictly rising order

Question

Three fair dice are rolled one after another. Find the probability that the values come out strictly increasing from the first roll to the third.

Short answer

$5/54$ . Any increasing triple is a choice of 3 distinct faces placed in their single sorted order, so there are $\binom63=20$ favourable outcomes out of $216$ , giving $\tfrac{20}{216}=\tfrac{5}{54}$ .

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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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