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A research team of 9 people consists of 4 people from the Boston office, 3 people from the Denver office, and 2 people from the Austin office. If a subcommittee of 4 people is selected at random from the 9 people, what is the probability that the subcommittee includes people from exactly two of the three offices?
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Correct answer
A
The sample space is choosing 4 from 9, which is C(9,4) = 126.
A subcommittee uses exactly two offices when all 4 members come from the union of two offices but not all from a single one. Handle each office pair, subtracting the all-one-office groups inside it.
Boston and Denver, 7 people: C(7,4) = 35, minus all-Boston C(4,4) = 1, minus all-Denver C(3,4) = 0, giving 34. Boston and Austin, 6 people: C(6,4) = 15, minus all-Boston 1, minus all-Austin C(2,4) = 0, giving 14. Denver and Austin, 5 people: C(5,4) = 5, minus all-Denver C(3,4) = 0, minus all-Austin C(2,4) = 0, giving 5.
So exactly two offices gives 34 + 14 + 5 = 53, and the probability is 53/126. The answer is A.
You can confirm this by completing the partition. Exactly one office: only all-Boston works, C(4,4) = 1, since Denver and Austin are each too small to fill 4 seats alone. Exactly three offices, a 2-1-1 split: C(4,2) × C(3,1) × C(2,1) = 36, plus C(4,1) × C(3,2) × C(2,1) = 24, plus C(4,1) × C(3,1) × C(2,2) = 12, for a total of 72. The buckets sum to 1 + 53 + 72 = 126, the full count.
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