Question: Committee of 2 Men & 2 Women

Comment on Committee of 2 Men & 2 Women

How is 6C2 =15
greenlight-admin's picture

nCr = n!/[r!(n-r)!]
So, 6C2 = 6!/[2!(6-2)!]
= 6!/[2!4!]
= (6)(5)(4)(3)(2)(1)/(2)(1)(4)(3)(2)(1)
= 15
Here's the video that first introduces combinations: https://www.greenlighttestprep.com/module/gre-counting/video/787

Here's a video that shows how to calculate combinations in your head: https://www.greenlighttestprep.com/module/gre-counting/video/789

I know the outcome of each stage do not differ from the outcome of the other, but is it possible to use the FCP with restrictions to solve this? I tried this method to no avail.
greenlight-admin's picture

We can still use the FCP, BUT we need to recognize when to use combinations within our solution.

This is what I have done in the video solution.

We took the task of creating the committee, and broke it into STAGES.

STAGE 1: Select the 2 men
STAGE 2: Select the 2 women

Aside: Notice that the outcomes of each stage are different. In one stage, we are selecting men and in the other stage, we are selecting women. So, we can still apply the FCP.

However, when we get to stage 1, we must recognize that IF we try to break that stage into smaller stages (e.g., select 1 man and then another man),the outcomes are the same. In both cases, we're selecting a man to be on the committee, which means we can't use the FCP. Instead, we must use combinations. That is, we can select 2 men from 6 men in 6C2 ways (15 ways). So, we can complete stage 1 in 15 ways.

The same goes for stage 2 (as in the video solution).

NOTE: If we were to break STAGES 1 and 2 into smaller stages, there is a strategy we COULD use to correct out calculations, but I'd rather not get into it, as it will cause more confusion than it will help.

Is this not a restriction question? If so, why? Please explain.
greenlight-admin's picture

I wouldn't classify questions as either restriction questions or non-restriction questions. That will lead to problems.

Also, we can pretty much classify all counting questions as restriction questions. Consider, for example, this question:

"Company X has 8 employees. In how many different ways can we select 3 employees to be on the Party-Planning committee?"

We COULD say that this is a restriction question, because we are restricted to having exactly 3 people on the committee.

Cheers,
Brent

Okay. Makes more sense now, after finishing the module :). Thanks Brent!

What is the explanation behind not being able to use FCP and do it as 6x5x4x3?
greenlight-admin's picture

As we try to set up the stage is to use in an FCP solution, we should ask, "Does the outcome of each stage differ from the outcomes of other stages?" (this is addressed in the following video: https://www.greenlighttestprep.com/module/gre-counting/video/788).

The FCP stages for the solution to the above problem would look something like this:

STAGE 1: Select a man to be on the committee
STAGE 2: Select another man to be on the committee
STAGE 3: Select a woman to be on the committee
STAGE 4: Select another woman to be on the committee

Notice that the outcomes for STAGES 1 and 2 are the same. In both cases, a MAN is selected to be on the committee. Since the outcomes are the same, we can't use the FCP.

The same applies to the outcomes for STAGES 3 and 4.

-----------------------------------
To show WHY we can't use the FCP, consider the exact same question, EXCEPT we only have 2 men and 2 women to choose from. This means, we must select all four people to be on the committee. So, there's only 1 possible outcome for this new question.

However, if we try to use the FCP, will get a different answer.

STAGE 1: Select a man to be on the committee: 2 men to choose from, so we can complete STAGE 1 in 2 ways.

STAGE 2: Select another man to be on the committee: We can complete the stage in 1 way.

STAGE 3: Select a woman to be on the committee: 2 women to choose from, so we can complete STAGE 2 in 2 ways.

STAGE 4: Select another woman to be on the committee: We can complete the stage in 1 way.

By the FCP, the answer = (2)(1)(2)(1) = 4, which is clearly incorrect.

Does that help?

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