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Company X has 6 regional offices. Each regional office must recommend

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Company X has 6 regional offices. Each regional office must recommend  [#permalink]

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New post Updated on: 25 Aug 2019, 21:12
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Company X has 6 regional offices. 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. 5
B. 10
C. 15
D. 20
E. 40

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Originally posted by vnigam21 on 05 Jun 2016, 12:46.
Last edited by Bunuel on 25 Aug 2019, 21:12, edited 2 times in total.
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Re: Company X has 6 regional offices. Each regional office must recommend  [#permalink]

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New post 05 Jun 2016, 21:41
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This is an easy question. I tried again and I was able to solve. :lol:

As the question says, If each of the offices must be represented by exactly one member on the auditing committee and there are 6 Regional Offices, so each of the regional offices will be represented by 6 members on the auditing committee.

Also, as per question, committee must consist of an equal number of male and female employees that is 3 Male members and 3 Female members. So, the question is now simply reduced to the possible number of arrangements for MMMBBB.

It can be done in 6!/3!3! ways that is equal to 20.
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Re: Company X has 6 regional offices. Each regional office must recommend  [#permalink]

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New post 05 Jun 2016, 23:06
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chetan2u Abhishek009, please tell me what m i missing here
Each office has two candidates M and F , so this way we got 6 M and 6 F
now question says only 1 M or F will be represent his/her office.
so my understanding says if M is selected than his counterpart F will not, and vice versa.
So 6M and 6F
so selection will be 6c3*3c3
where 6c3 represents 3 males out of 6 and 3c3 represents remaining 3 females , which are not male counterpart.
similarly we have case for female 6c3 * 3c3
so ultimatly we 2 6c3= 40.
Please correct me where I am wrong.
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Re: Company X has 6 regional offices. Each regional office must recommend  [#permalink]

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New post 06 Jun 2016, 03:18
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sudhirmadaan wrote:
chetan2u Abhishek009, please tell me what m i missing here
Each office has two candidates M and F , so this way we got 6 M and 6 F
now question says only 1 M or F will be represent his/her office.
so my understanding says if M is selected than his counterpart F will not, and vice versa.
So 6M and 6F
so selection will be 6c3*3c3
where 6c3 represents 3 males out of 6 and 3c3 represents remaining 3 females , which are not male counterpart.
similarly we have case for female 6c3 * 3c3
so ultimatly we 2 6c3= 40.
Please correct me where I am wrong.


Hi,

the method or formula you are working on, is not correct here...

when you make a group of 3 from 6, the other group is automatically made as only 3 are left after choosing initial 3, and the same will be applicable here ..

Ways to divide 2 groups of 3 each from 6 members....
say ABCDEF are 6 persons and you choose ABC in 6C3, another group DEF is automaticaly formed..
so when you choose DEF in 6C3, ABC is also formed and thus there is a repetition and that is why divison by 2!..
ans 6C3/2!.....
But as you have found 2 * 6C3, which ACTUALLY should be\(2*\frac{6C3}{2!} = 6C3.\)....

SOLUTION : Another way :-

Let the 6 companies be ABCDEF..
Now the 3 we choose in 6C3 send male and the remaining 3 will send female....
so we basically find HOW many ways 2 teams of 3 each can be choosen out of 6 = 6!/3!3!2!= 10..

But each team will once send MALE and other time FEMALE ...
so ans = 10*2 = 20
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Re: Company X has 6 regional offices. Each regional office must recommend  [#permalink]

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New post 25 Jan 2019, 04:18
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if each office can recommend 2 members, one male and one female, the potential members of the committee are 12, 6 males and 6 females. Since each office will be represented by 1 person and the committee must have an equal number of males and females, the committee will be formed by exactly 6 people, 3 males and 3 females.
Now we have to select 3 males in a group of 6 males and 3 females in a group of 6 females. Therefore, we use the combination formula: 6!/(3!*3!) -> number of ways of choosing either 3 males out of 6, or 3 females out of 6; and then this result must be squared, to multiply all the possible combinations of males and females.
Finally, the result is: 400
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Re: Company X has 6 regional offices. Each regional office must recommend  [#permalink]

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New post 25 Aug 2019, 17:22
the committee must consist of 6 members. Further, because the committee must have an equal number of male and female employees, it must include 3 men and 3 women. First, let's form the female group of the committee. There are 3 women to be selected from 6 female candidates (one per region). One possible team selection can be represented as follows, where A, B, C, D, E, & F represent the 6 female candidates:
A B C D E F
Yes Yes Yes No No No
In the representation above, women A, B, and C are on the committee, while women D, E, and F are not. There are many other possible 3 women teams. Using the combination formula, the number of different combinations of three female committee members is 6! / (3! × 3!) = 720/36 = 20. To ensure that each region is represented by exactly one candidate, the group of men must be selected from the remaining three regions that are not represented by female employees. In other words, three of the regions have been “used up” in our selection of the female candidates. Since we have only 3 male candidates remaining (one for each of the three remaining regions), there is only one possible combination of 3 male employees for the committee. Thus, we have 20 possible groups of three females and 1 possible group of three males for a total of 20 × 1 = 20 possible groups of six committee members.
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Re: Company X has 6 regional offices. Each regional office must recommend   [#permalink] 25 Aug 2019, 17:22
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