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?
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Answer & Explanation
Correct answer
A
If the 3 groups were labeled Group 1, Group 2, and Group 3, the count would be C(9,3) × C(6,3) × C(3,3) = 84 × 20 × 1 = 1680. Choose 3 of the 9 students for Group 1, then 3 of the remaining 6 for Group 2, then the last 3 form Group 3.
But the groups are unlabeled and interchangeable, so each unlabeled division is counted 3! = 6 times, once for every way to attach the labels 1, 2, 3 to the same 3 groups. Divide out this overcount: 1680 ÷ 6 = 280.
As an independent check, use the multinomial-over-symmetry formula directly: 9!/((3!)³ × 3!) = 362880/(216 × 6) = 362880/1296 = 280, which matches.