GMATPrep Practice Exam: Combinations

[talueng]

A 4-person task force is to be formed from the 4 men and 3 women who work in Company G's human resources department. If there are to be 2 men and 2 women on this task force, how many different task forces can be formed?

a. 14
b. 18
c. 35
d. 56
e. 144

My answer would be that n = 4*3*3*2, but 72 does not exist as a possible answer. GMATPrep marks b: 14, which obviously is 4*2+3*2, as the correct answer, and I do not understand how this can be the correct answer (say: I disagree).

I have 4 positions available in the team. The first can be taken by one of the 4 men, the second by one of the remaining 3 men, the third by one of the 3 women, and the fourth by one of the remaining 2 women, thus 4*3*3*2.
Even if I break the problem up into two smaller problems, asking how many different men task forces and women task forces I can build, which, as it seems, the GMATPrep answer does, I do not understand why the resulting combinations would be added up rather than multiplied. I can build 4*3 different men task forces, and 3*2 different women task forces, and combining each of them I can build 12*6 different mixed task forces.

Please help clarifying the answer.

[GMATGuruNY]

A 4-person task force is to be formed from the 4 men and 3 women who work in Company G's human resources department. If there are to be 2 men and 2 women on this task force, how many different task forces can be formed?

a. 14
b. 18
c. 35
d. 56
e. 144

My answer would be that n = 4*3*3*2, but 72 does not exist as a possible answer. GMATPrep marks b: 14, which obviously is 4*2+3*2, as the correct answer, and I do not understand how this can be the correct answer (say: I disagree).

I have 4 positions available in the team. The first can be taken by one of the 4 men, the second by one of the remaining 3 men, the third by one of the 3 women, and the fourth by one of the remaining 2 women, thus 4*3*3*2.
Even if I break the problem up into two smaller problems, asking how many different men task forces and women task forces I can build, which, as it seems, the GMATPrep answer does, I do not understand why the resulting combinations would be added up rather than multiplied. I can build 4*3 different men task forces, and 3*2 different women task forces, and combining each of them I can build 12*6 different mixed task forces.

Please help clarifying the answer.

Since the ORDER of the people chosen doesn't matter -- selecting AB for the task force is the same as selecting BA -- we must DIVIDE by the number of ways the selected people can be ARRANGED.

Number of ways to choose 2 men from 4 options = (4*3)/(2*1) = 6.
Number of ways to choose 2 women from 3 options = (3*2)/(2*1) = 3.
To combine these options, we multiply:
6*3 = 18.

The correct answer is B.

[Brent@GMATPrepNow]

A 4-person task force is to be formed from the 4 men and 3 women who work in Company G's human resources department. If there are to be 2 men and 2 women on this task force, how many different task forces can be formed?
a. 14
b. 18
c. 35
d. 56
e. 144

We can take the task of "building" a task force and break it into stages.

Stage 1: select the 2 women for the team
Since the order of the selected women does not matter, this is a combination question.
There are 3 women, and we want to choose 2 of them.
This can be accomplished in 3C2 ways, which equals 3

Stage 2: select the 2 men for the team
Since the order of the selected men does not matter, this is a combination question.
There are 4 men, and we want to choose 2 of them.
This can be accomplished in 4C2 ways, which equals 6

By the Fundamental Counting Principle (FCP) we can complete both stages (and create the committee) in (3)(6) ways = 18 ways

So the answer is B

Cheers,
Brent

Aside: If anyone is interested, we have a free video on calculating combinations (like 4C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789