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Every face appears

ProbabilityHard

A fair $6$ -sided die is rolled $8$ times, all rolls independent. What is the probability that every face from $1$ to $6$ appears at least once? Give a fraction in lowest terms.

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Hint

Count outcomes where some faces are missing with inclusion-exclusion over the set of missing faces.

Solution

There are $6^{8} = 1679616$ equally likely outcomes.
Outcomes avoiding a given set of $k$ faces number $(6 - k)^{8}$ .
Inclusion-exclusion gives $1679616 - 6 \times 390625 + 15 \times 65536 - 20 \times 6561 + 15 \times 256 - 6 \times 1$ .
This equals $1679616 - 2343750 + 983040 - 131220 + 3840 - 6 = 191520$ .
The probability is $\frac{191520}{1679616}$ , which reduces to $\frac{665}{5832}$ .

Answer: $\frac{665}{5832}$

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