Nine students are split into three equal teams to develop reports on

Question

Nine students are split into three equal teams to develop reports on one of three problems: shortage of skilled labor, violence in schools, and low standardized test scores. If each team will report on a different problem, then how many different assignments of students to problems are possible?

(A) 5040
(B) 1680
(C) 1512
(D) 504
(E) 168

(A) 5040
(B) 1680
(C) 1512
(D) 504
(E) 168

Vivor · Answer

Problem I = shortage of skilled labor; Problem II = violence in schools; Problem III = low standardized test scores

Since 3 equal teams are formed from 9 students, no. of students in each team = 3

No. of ways in which 3 (out of 9) students can be chosen for Problem I = 9C3
No. of ways in which 3 (out of remaining 6) students can be chosen for Problem II = 6C3
No. of ways in which 3 (out of remaining 3) students can be chosen for Problem III = 3C3

Hence, total no. of ways = 9C3 * 6C3 * 3C3 = 1680

Ans is B IMO

twobagels · Answer

9!/(3! * 3! * 3!) = 1,680

ScottTargetTestPrep · Answer

Solution:
 
First, there are   / 3! =   / 6 =   / 6 = 280 ways the split 9 people into 3 equal teams. (Notice that 9C3 is the number of ways the first team could be chosen, 6C3 the second, and 3C3 the third. But once 3 teams are chosen, it doesn’t matter how we arrange them, they are still the same 3 teams; therefore, we need to divide by 3!).

Second, for every set of 3 teams, there are 3! = 6 ways the 3 reports can be assigned to them. Therefore, there are a total of 280 x 6 = 1680 different assignments.

Alternate Solution:
 
For the team reporting on shortage of skilled labor, there are 9C3 = 9!/(3! * 6!) = 84 choices.

Assuming the team reporting on shortage of skilled labor is chosen, there are 6C3 = 6!/(3! * 3!) = 20 choices for the team reporting on violence in schools.

If the teams reporting on shortage of skilled labor and violence in schools are chosen, then the three remaining students must form the team reporting on low standardized test scores; thus there is only one choice for that team.

By the fundamental principle of counting, there are 84 x 20 = 1680 ways to choose the teams and assign to a problem.

Answer: B

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Nine students are split into three equal teams to develop reports on

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