Question
Company X has n regional offices, where n represents an even integer. Each regional office must recommend two candidates, one male and one female, to serve on the corporate auditing committee. If each of the offices must be represented by exactly one member on the auditing committee and if the committee must consist of an equal number of male and female employees, how many different committees can be formed?

(A) Square of (.5n)
(B) n!/(Square of ((.5n)!))
(C) Square of (n!/((0.5n)!(.5n)!)))
(D) n!/((0.5n)!)
(E) n!

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Ans is (B)

I got (C) using the following reasoning -

Since each office recommends 1 man and 1 woman, we totally have n men and n women to choose a total of n poeople from (comprising of n/2 men and n/2 women)

No of ways of selecting the men = n C n/2 --(1)

No of ways of selecting the women = n C n/2 -- (2)

Multiplying (1) and (2), I got answer choice (C). Could you please explain as to where I may be going wrong here?

Thanks!

Each regional office has 1M 1F
Total number of regional offices is n
Total number of males available is n males
Total number of females available is n females
Need to select equal number of males and females..n is even...n/2 males and n/2 females

Combination of selecting n/2 males from n males and n/2 females from n females .

Therefore n C n/2 . n C n/2 ..ans I am getting C


e.g take four regional members
Male : M1, M2, M3, M4 - fem: F1,F2, F3 &F4

total number of arrangements
M1M2F1F2
M1M2F1F3
M1M2F1F4
M1M2F2F3
M1M2F2F4
M1M2F3F4

M1M3F1F2
M1M3F1F3
M1M3F1F4
M1M3F2F3
M1M3F2F4
M1M3F3F4
....total = 36ways

B gives only 6 ways

Hi Mandy,

Let's just focus on the women (or the men, if you prefer). If I've already chosen the n/2 women who will be on the team, there's no mystery at all about which men will be on the team - they'll be the men from the n/2 offices that are not sending a woman.

For example, if there are offices in Atlanta, Boston, Chicago, and Denver, and you've chosen to take women from Boston and Denver, the men are simply going to have to come from Atlanta and Chicago.

So to count the total number of teams possible, I need only count the ways of choosing the n/2 women on the team.

As you mentioned in your posting, the number of ways of choosing n/2 women out of a total of n women is just n C n/2. So the answer to the original question is n C n/2. And what is n C n/2 ? Using the formula for combinations, it's simply n!/((n/2)!(n-n/2)!), which simplifies to answer choice B. (Remember that (n-n/2) =n/2.)

-Jad

Hi Dred,

When you're counting combinations or permutations, it's important to remember not to count combinations or permutations that violate the constraints given in the problem.

You must not multiply the (number of ways of selecting n/2 males from n males) times (the number of ways of selecting n/2 females from n females). If you have already selected the n/2 males, you simply have no choice left as to which females you must take. For example, if there are offices in Atlanta, Boston, Chicago, and Denver, and you've chosen to take men from Boston and Denver, the women are going to have to come from Atlanta and Chicago. Once you've chosen the men, there is only one way of choosing the women. (For more on this logic, see my answer to Mandy's post above.)

In your list of possibilities below, all of the ones I've highlighted are impossible, because each office can send only one person. For example, in M1M2F1F2 you have two people from office 1 (M1 and F1) and two people from office 2 (M2 and F2) and nobody from office 3 or 4.

-Jad

Thanks Jadran. That was a really good explanation.

Thanks much Jadran. Your explanation makes a lot of sense.

Have a great weekend!

-Mandy