PrepLattice is an independent test-preparation service and is not affiliated with or endorsed by GMAC, the organization that administers the GMAT. GMAT and GMAT Focus are trademarks of GMAC, used here only to name the exam this question is designed to prepare you for.
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?
Five fresh questions every day, your progress tracked, every miss explained. Free with an account.
Correct answer
B
Total ways to draw 4 marbles from 9: C(9,4) = 126.
Exactly two colors means all four marbles come from exactly two of the three color pools, with both colors actually appearing. Take each pair of colors, count the ways to choose 4 from the combined pool, then subtract the selections that use only one color. Red and white pool, 7: 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: C(6,4) = 15, minus all-red 1, minus all-blue C(2,4) = 0, gives 14. White and blue pool, 5: C(5,4) = 5, minus 0, minus 0, gives 5. Sum = 34 + 14 + 5 = 53, so the probability is 53/126. The answer is B.
A complement check confirms it. All one color: only red can supply 4, C(4,4) = 1. All three colors, a 2-1-1 split: red gives 2, C(4,2)×C(3,1)×C(2,1) = 36; white gives 2, C(3,2)×C(4,1)×C(2,1) = 24; blue gives 2, C(2,2)×C(4,1)×C(3,1) = 12; sum 72. Then exactly two colors = 126 − 1 − 72 = 53.
The distractors track specific errors. 55/126 (C) is the dominant trap, adding the pair counts (35 + 15 + 5) without removing the all-one-color draws inside each pair. 17/63 (A) keeps only the largest pair (34/126) and forgets the other two. 4/7 (D) is the all-three-colors probability, the complementary case. 125/126 (E) answers at least two colors instead of exactly two.
Pick the theme that suits how you work. You can change it anytime.