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Indistinguishable dice outcomes

CombinatoricsEasy

You roll $5$ identical $6$ -sided dice that cannot be told apart. How many distinct outcomes, counted as multisets of faces, can appear?

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Hint

An outcome is just how many dice show each face.

Solution

An outcome is fixed by the counts $n 1 + n 2 + n 3 + n 4 + n 5 + n 6 = 5$ with $ni \geq 0$ .
Stars and bars gives $(5 5 + 5)$ .
Compute $(5 10) = 252$ .
The number of outcomes is $252$ .

Answer: $252$