Solution
Four Identical Dice
Show the problem again
Four indistinguishable standard six-sided dice are rolled. An outcome is recorded only as the multiset of faces showing (for example, {1, 1, 4, 6}), since the dice cannot be told apart. How many different outcomes are possible? Give an integer.
Worked solution
The answer is 126. An outcome is determined by how many dice show each face, that is, by nonnegative integers n₁, ..., n₆ with n₁ + ... + n₆ = 4. By stars and bars (4 stars, 5 bars) the count is C(9, 4) = 126. Note this differs sharply from the 6⁴ = 1296 ordered outcomes, and the multisets are not equally likely.
Source: Classic counting exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.