Six rolls, all six faces

Question

You roll a fair six-sided die 6 times in a row. What is the probability that all six faces (1 through 6) each appear exactly once?

Accepted Answer

There are 6^6 equally likely outcome sequences, since each of the 6 rolls has 6 possibilities: 6^6 = 46{,}656. For every face to appear exactly once, the sequence of results must be a rearrangement (permutation) of \{1,2,3,4,5,6\}. The number of such orderings is 6! = 720. So P = \frac{6!}{6^6} = \frac{720}{46{,}656} = \frac{5}{324} \approx 0.0154. Only about 1.5% of the time do six rolls give a perfect sweep of all faces. Takeaway. "Each outcome exactly once in n trials of n options" is \frac{n!}{n^n}, favorable orderings over all sequences. This ratio shrinks fast with n (it's essentially e…

Six rolls, all six faces

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