Janice has divided her department of 12 employees into 4 teams of 3

Question

Janice has divided her department of 12 employees into 4 teams of 3 employees each to work on a new project. Four new employees are to be added to Janice’s department, but the number of teams is to remain the same. If every employee must be on exactly 1 team, then the total number of different teams into which Janice can divide the enlarged department is approximately how many times the number of different teams into which Janice could have divided the original department?

A 16
B 170
C 340
D 1820
E 43,680

IanStewart · Answer

The wording of this question is hopeless. The question absolutely needs to tell you that the enlarged department will be divided into four teams  of four employees each . The question doesn't tell you anything about how big the teams will be in the enlarged department, and if you assume they can be of any size (which is what I would assume from the wording of the question), the correct answer is much bigger than 170 (and the problem would then be much too complicated to solve in 2 minutes).

If we assume the four teams must always be of equal size, then when we pick the four teams in the 12-person department, we can pick the first team in 12C3 ways, the second in 9C3 ways, the third in 6C3 ways, and the fourth in 3C3 ways. So if we assume we're picking teams in order, we can pick them in:

12C3 * 9C3 * 6C3 * 3C3 = 12! / (3!)^4

ways. The order of the 4 teams presumably doesn't matter, so we'd divide this by 4! if this were a counting problem, but we'll also divide by 4! when we count how many teams we can make in the larger department, and those two 4!'s will cancel in the end anyway, so I'll just omit that term.

When we have 16 employees, and divide them into four teams of four, we can similarly do that in

16C4 * 12C4 * 8C4 * 4C4 = 16! / (4!)^4

ways if the order of the teams matters. We need to divide this by the result above, so we get:

\dfrac{ \frac{16!}{(4!)^4} }{ \frac{12!}{(3!)^4} } = \frac{(16!)(3!)^4}{(12!)(4!)^4} = \frac{(16)(15)(14)(13)}{4^4} = \frac{(15)(14)(13)}{4^2}

which is roughly equal to 170.

I've never seen a real GMAT question that requires you to do anything like this, so anyone who finds this question difficult really shouldn't be concerned.

Janice has divided her department of 12 employees into 4 teams of 3

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