Consider an urn with 30 beads--10 each of red, blue, and green--and randomly pull these beads out in groups of 3, as shown.
What is the probability that each of the 10 possible color combinations is represented?
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3 0 ! 1 0 ! ∗ 1 0 ! ∗ 1 0 ! ∗ 3 6 ∗ 6 ∗ 1 0 !
Each of the first three 1 0 ! represents fact that it doesn't matter which red (or other) bead is pulled in each trio. The 3 6 is because the ordering of each trio with two of one color is unimportant. The six is for the orders of the trio with one of each color. The final 10! is because the order in which the trios was pulled is unimportant.
This fraction reduces to 4 1 0 5 5 5 3 0 0 9 1 2 5 9 7 1 2 0 ≈ 0 . 0 0 2 8 6