A class of 9 students is to be divided into 3 unlabeled groups of 3 students each for a project. In how many distinct ways can this division be made?
Five fresh questions every day, your progress tracked, every miss explained. Free with an account.
QuantProblem Solving
Free GMAT Problem Solving Practice Question
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.
Answer & Explanation
Correct answer
A
If the 3 groups were labeled Group 1, 2, and 3, the count would be C(9,3) × C(6,3) × C(3,3) = 84 × 20 × 1 = 1680. But the groups are unlabeled and interchangeable, so each unlabeled division is counted 3! = 6 times, once for every way to attach labels to the same 3 groups. Divide: 1680/6 = 280.
Verification by the multinomial-over-symmetry formula: 9!/((3!)³ × 3!) = 362880/(216 × 6) = 362880/1296 = 280, which matches.
Choice (B) 840 = 1680/2 divides by only 2!, (C) 1680 leaves the groups labeled, and (E) 362880 = 9! orders all students.