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A jar holds 4 red marbles, 3 white marbles, and 2 blue marbles, all distinguishable. Four marbles are drawn from the jar at random, all at once. What is the probability that the four marbles drawn come from exactly two of the three colors?
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Correct answer
B
There are C(9,4) = 126 ways to draw 4 marbles from the 9 in the jar.
Exactly two colors means all four marbles come from exactly two of the three color pools, with both of those colors actually appearing among the four. Take each pair of colors, count the ways to choose 4 from that combined pool, then subtract the selections that use only one color from the pair.
Red and white pool, 7 marbles: C(7,4) = 35, minus all-red C(4,4) = 1, minus all-white C(3,4) = 0, gives 34. Red and blue pool, 6 marbles: C(6,4) = 15, minus all-red 1, minus all-blue C(2,4) = 0, gives 14. White and blue pool, 5 marbles: C(5,4) = 5, minus all-white 0, minus all-blue 0, gives 5.
Add the three cases: 34 + 14 + 5 = 53, so the probability is 53/126. The answer is B.
A complement check confirms this. All one color: only red has enough marbles to supply all 4, so C(4,4) = 1. All three colors, which must split 2-1-1 among the four marbles: 2 red gives C(4,2) × C(3,1) × C(2,1) = 36; 2 white gives C(3,2) × C(4,1) × C(2,1) = 24; 2 blue gives C(2,2) × C(4,1) × C(3,1) = 12; these sum to 72. Then exactly two colors = 126 − 1 − 72 = 53, matching the direct count.
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