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# A committee of three people is to be chosen from four married couples.

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A committee of three people is to be chosen from four married couples.  [#permalink]

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Updated on: 16 Sep 2019, 06:26
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A committee of three people is to be chosen from four married couples. What is the number of different committees that can be chosen if two people who are married to each other cannot both serve on the committee?

A. 16
B. 24
C. 26
D. 30
E. 32

Approach I thought is as follows...if some shorter method is possible please explain..

total selections = 8C3 = 56

let's say that couple is always present in this committee of three.

This means that there are 4 ways to select 2 people of the committee. ( 4 couples and any one couple can be selected in 4 ways)
The third person can be selected out of remaining 6 people in 6 ways.

Therefore when couple exists there are: 4X6 = 24 ways

Thus no couple = 8C3 - (4X6) = 32

Originally posted by LM on 11 May 2010, 11:35.
Last edited by Bunuel on 16 Sep 2019, 06:26, edited 3 times in total.
Edited the question and added the OA
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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11 May 2010, 14:24
46
73
A committee of 3 people is to be chosen from four married couples. What is the number of different committees that can be chosen if two people who are married to each other cannot both serve on the committee?
A. 16
B. 24
C. 26
D. 30
E. 32

One of the approaches:

Each couple can send only one "representative" to the committee. Let's see in how many ways we can choose 3 couples (as there should be 3 members) out of 4 to send only one "representatives" to the committee: 4C3=4.

But each of these 3 couples can send two persons (husband or wife): 2*2*2=2^3=8.

Total # of ways: 4C3*2^3=32.

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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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29 Oct 2010, 01:28
123
1
33
Since there are 4 couples so we have 8 people involved.

The First person can be selected from the 8 people in 8 ways
The second person should not be a spouse of the first and hence we have 6 ways to choose him/her
The Third person should not be a spouse of either of the 2, so we can choose him in 6 ways.

So the total no. of ways we can choose the people will be 8*6*4 ways.
However since order is not important (i.e A,B,C is the same as B,A,C) so we divide the total ways by 3!

Hence the total number of groups is 32
##### General Discussion
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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17 Jun 2010, 04:50
31
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amitjash wrote:
Hi everybody,
I want to solve this problem with some other method. But i am definately wrong somewhere in this method... i dont understand where.

i can choose first member of the comitee in 8 ways..
removing the spouse of the selected person second member can be chosen in 6 ways....
third member in 4 ways.....
so 8*6*4 which is not answer can someone explain why?

The way you are doing is wrong because 8*6*4=192 will contain duplication and to get rid of them you should divide this number by the factorial of the # of people - 3! --> 192/3!=32.

Consider this: there are two couples and we want to choose 2 people not married to each other.
Couples: $$A_1$$, $$A_2$$ and $$B_1$$, $$B_2$$. Committees possible:

$$A_1,B_1$$;
$$A_1,B_2$$;
$$A_2,B_1$$;
$$A_2,B_2$$.

Only 4 such committees are possible.

If we do the way you are doing we'll get: 4*2=8. And to get the right answer we should divide 8 by 2! --> 8/2!=4.

You can see the similar problem at:
committee-of-88772.html#p669797

Hope it helps.
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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28 Oct 2010, 21:45
21
9
So there are 3 people we need to choose from 8.

Case 1 : all 3 are men ... C(4,3)=4 ways
Case 2 : all 3 are women ... C(4,3)=4 ways
Case 3 : 2 men 1 woman ... Choose men in C(4,2) ways, then we can only choose the woman in 2 ways, since their wives can't be chosen .., hence 12x2=24 ways
Case 4 : 2 women 1 man ... Exactly same logic as case 3, 12 ways

Total ways = 4+4+12+12 = 32

Posted from my mobile device
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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17 Jun 2010, 01:16
5
Hi everybody,
I want to solve this problem with some other method. But i am definately wrong somewhere in this method... i dont understand where.

i can choose first member of the comitee in 8 ways..
removing the spouse of the selected person second member can be chosen in 6 ways....
third member in 4 ways.....
so 8*6*4 which is not answer can someone explain why?
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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06 Aug 2012, 10:21
4
Person (p1 p2 p3 p4 p5 p6 p7 p8)
No of ways to choose 1st Person: Any 8
No of ways to choose 2nd Person: 6 (Pair of 1st person can not be considered so we need to exclude 1 pair)
No of ways to choose 3rd Person: 4 (Pair of 1st & 2nd Person can not be considered so we need to exclude 2 pair)
No of ways : 8X6X4 (Now we have done a permutation)
But here order of the team member is not important and 3 person can arrange themselves in 3! ways. So need to divide the permutation by 3!.
Ans: 8*6*4/3! = 32
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Posts: 65764
Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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07 Nov 2012, 04:37
4
5
watwazdaquestion wrote:
is this a correct way to get the answer? or was it just coincidence:

8C3 - 4(4C2) = 56 - 4(6) = 32

It's not clear what is the logic behind the formula.

Reversed approach would be:
There are 8C3=56 ways to select 3 people out of 8 without any restriction;
There are 4C1*6=24 ways there to be a couple among 3 members: 4C1 ways to select a couple out of 4, which will be in the committee and 6 ways to select the third remaining member (since there will be 6 members left after we select a couple out of 8 people).

56-24=32.

Hope it's clear.
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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12 Jan 2018, 13:52
2
4
Hi All,

This question is fairly high-concept, but the math behind it isn't too complex. You have to be organized and you have to remember that we're forming GROUPS of people, so 'duplicate entries' are not allowed.

From the prompt, we have 4 married couples, but we have to form a group of 3 without putting a married couple in the group. Let's work through the logic in pieces...

The 1st person in the group can be ANY of the 8 people. Once we pick one of those 8, then we CANNOT pick the married partner to that person...

For the 2nd person in the group, we now have 6 people to choose from. Once we pick one of those 6, then we CANNOT pick the married partner to that person...

For the 3rd person in the group, we now have 4 people to choose from.

We now multiply those three numbers together: (8)(6)(4) = 192

We're NOT done though. Remember that we're forming GROUPS of people, and the 192 options we've figured out so far contain LOTS of duplicates... If you have 3 people: A, B and C, then that is just ONE group. However, in the above approach, you can end up with that group in 6 different ways:

ABC
ACB
BAC
BCA
CAB
CBA

Thus, every actual option has been counted 6 times, so we have to divide that total by 6...

192/6 = 32

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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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29 Oct 2019, 01:43
2
Choosing 3 out of 8 can be done in 8C3= 56 ways. From this we need to remove the ones which have couples who are married to each other. Let's say the couples are (M1F1) (M2 F2) (M3F3) and (M4F4).
If we choose M1F1 then there are 6 choices for the third member. Similarly there will be 6 cases in which M2F2 will be there and 6 in which M3F3 will be present and 6 in which M4F4 will be present. That makes it 24 sets in which couples who are married to each other would be present. So subtracting this from 56 we have 32 sets or groups in which no couple would be present. Hence E

Posted from my mobile device
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Posts: 65764
Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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24 Apr 2014, 00:25
1
1
russ9 wrote:
Bunuel wrote:
watwazdaquestion wrote:
is this a correct way to get the answer? or was it just coincidence:

8C3 - 4(4C2) = 56 - 4(6) = 32

It's not clear what is the logic behind the formula.

Reversed approach would be:
There are 8C3=56 ways to select 3 people out of 8 without any restriction;
There are 4C1*6=24 ways there to be a couple among 3 members: 4C1 ways to select a couple out of 4, which will be in the committee and 6 ways to select the third remaining member (since there will be 6 members left after we select a couple out of 8 people).

56-24=32.

Hope it's clear.

Hi Bunuel,

I'm confused by this step, which is also outlined above. "There are 4C1*6=24 ways there to be a couple among 3 members:". Can you please elaborate on this?

First couple: $$A_1,A_2$$;
Second couple: $$B_1,B_2$$;
Third couple: $$C_1,C_2$$;
Fourth couple: $$D_1,D_2$$.

We want to select 3 people: a couple and one more.

We can select any from 4 couples (4 options) and for the third member we can select any from the remaining 6 people. For example if we select $$A_1,A_2$$, then we can select third member from $$B_1,B_2,C_1,C_2,D_1,D_2$$.

Hope it's clear.
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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25 Jun 2016, 15:37
1
1
We can solve this in two general steps.

1) We can pretend order matters (which can be much easier to visualize)
2) Then we can think about what happens if order doesn't matter.

1) If order matters, we can visualize three chairs or slots or whatever and consider them one at a time: Anybody can sit in the first chair, so there are 8 options for chair one. But now that somebody is in chair one, there are only 6 options for chair two (it can't be the person in chair one and it can't be that person's spouce). And for every set of people occupying chairs one and two, there are only 4 remaining people allowed to sit in chair three.

That means that there are 8*6*4 ways to fill the three chairs.

2) But we're looking for 'committees.' and for committees order doesn't matter (whereas it does when we're visualizing chairs). For example, Andy, Beth, Charlie would be the same committee as Beth, Andy Charlie.

In fact there are six ways that Andy Beth and Charlie could fill the chairs:

ABC
ACB
BAC
BCA
CAB
CBA

So for each committee, we counted 6 different ways for them to fill the chairs.

In other words, our 8*6*4 number is too big by a factor of 6.

Therefore, the answer is (8*6*4)/6 = 32
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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06 Nov 2012, 11:53
is this a correct way to get the answer? or was it just coincidence:

8C3 - 4(4C2) = 56 - 4(6) = 32
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Posts: 65764
A committee of three people is to be chosen from four married couples.  [#permalink]

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27 Oct 2013, 04:47
schokshi99 wrote:
A committee of three people is to be chosen from four married couples. What is the number of different committees that can be chosen if two people who are married to each other cannot both serve on the committee?

A. 16
B. 24
C. 26
D. 30
E. 32

Similar problems to pracitce:
http://gmatclub.com/forum/a-committee-o ... 30617.html
http://gmatclub.com/forum/if-4-people-a ... 99055.html
http://gmatclub.com/forum/a-committee-o ... 94068.html
http://gmatclub.com/forum/if-a-committe ... 88772.html
http://gmatclub.com/forum/a-comittee-of ... 30475.html
http://gmatclub.com/forum/a-committee-o ... 01784.html
http://gmatclub.com/forum/a-group-of-10 ... 13785.html
http://gmatclub.com/forum/if-there-are- ... 99992.html
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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23 Apr 2014, 19:31
Bunuel wrote:
watwazdaquestion wrote:
is this a correct way to get the answer? or was it just coincidence:

8C3 - 4(4C2) = 56 - 4(6) = 32

It's not clear what is the logic behind the formula.

Reversed approach would be:
There are 8C3=56 ways to select 3 people out of 8 without any restriction;
There are 4C1*6=24 ways there to be a couple among 3 members: 4C1 ways to select a couple out of 4, which will be in the committee and 6 ways to select the third remaining member (since there will be 6 members left after we select a couple out of 8 people).

56-24=32.

Hope it's clear.

Hi Bunuel,

I'm confused by this step, which is also outlined above. "There are 4C1*6=24 ways there to be a couple among 3 members:". Can you please elaborate on this?
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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20 May 2017, 19:35
Top Contributor
LM wrote:
A committee of 3 people is to be chosen from four married couples. What is the number of different committees that can be chosen if two people who are married to each other cannot both serve on the committee?

A. 16
B. 24
C. 26
D. 30
E. 32

Approach I thought is as follows...if some shorter method is possible please explain..

total selections = 8C3 = 56

let's say that couple is always present in this committee of three.

This means that there are 4 ways to select 2 people of the committee. ( 4 couples and any one couple can be selected in 4 ways)
The third person can be selected out of remaining 6 people in 6 ways.

Therefore when couple exists there are: 4X6 = 24 ways

Thus no couple = 8C3 - (4X6) = 32

1. Order is not important and there is no repetition , so this is an nCr problem.
2. Is there a constraint? Married couple cannot be in the same committee.
3. It is easier to find the opposite of the constraint
4. Total number of combinations is 8C3
5. 1 married couple can be there in the 3 selected for a total of 4 such cases since there are 4 married couples. The remaining 1 person can be selected in 6 ways, for a total of 4*6=24 ways
6. Total number of combination with constraints is (4)-(5)=32
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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13 Dec 2017, 17:27
LM wrote:
A committee of 3 people is to be chosen from four married couples. What is the number of different committees that can be chosen if two people who are married to each other cannot both serve on the committee?

A. 16
B. 24
C. 26
D. 30
E. 32

We are given that there are 4 married couples (or 8 people), and we need to determine the number of ways of choosing 3 people in which no 2 people are a married couple. So this is a special combination problem. Before we tackle this problem, let’s review a combination problem with no restrictions.

With no restrictions, the number of ways of choosing 3 people from 8 is 8C3, which is calculated as follows:

8C3 = 8!/[3!(8-3)!] = (8 x 7 x 6)/3! = 56

8, 7, and 6 in the numerator represent the number of ways in which the first, second, and third person can be chosen, respectively. We divide the numerator by 3! because, in a combination problem, we do not care about the order in which the 3 people are chosen.

However, in this (special combination) problem, 3 people are chosen in which no married couple can serve together on the committee. The first person could be any one of the 8 people. However, once a person is selected, that person’s spouse cannot also be selected for the committee. This reduces the choice of the second person to 6 possible people (one person has already been selected and that person’s spouse now cannot be selected). Once the second person is chosen for the committee, that person’s spouse cannot be chosen. This reduces the number of people who could be chosen as the third person to 4. Therefore, the number of ways of choosing these 3 people is:

(8 x 6 x 4)/3! = 32

Thus, there are 32 ways to choose such a committee.

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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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13 Mar 2018, 13:32
Top Contributor
1
LM wrote:
A committee of 3 people is to be chosen from four married couples. What is the number of different committees that can be chosen if two people who are married to each other cannot both serve on the committee?

A. 16
B. 24
C. 26
D. 30
E. 32

Take the task of selecting the 3 committee members and break it into stages.

Stage 1: Select the 3 couples from which we will select 1 spouse each.
There are 4 couples, and we must select 3 of them. Since the order in which we select the 3 couples does not matter, we can use COMBINATIONS
We can select 3 couples from 4 couples in 4C3 ways (4 ways)

Stage 2: Take one of the 3 selected couples and choose 1 person to be on the committee.
There are 2 people in the couple, so this stage can be accomplished in 2 ways.

Stage 3: Take one of the 3 selected couples and choose 1 person to be on the committee.
There are 2 people in the couple, so this stage can be accomplished in 2 ways.

Stage 4: Take one of the 3 selected couples and choose 1 person to be on the committee.
There are 2 people in the couple, so this stage can be accomplished in 2 ways.

By the Fundamental Counting Principle (FCP) we can complete all 4 stages (and thus create a 3-person committee) in (4)(2)(2)(2) ways (= 32 ways)

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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12 May 2018, 16:09
You are trying to form a committee of six people at work to run a group project. If you have fifteen coworkers to choose from, how many different committees can you form?
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Re: A committee of three people is to be chosen from four married couples.  [#permalink]

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12 May 2018, 23:47
marycosme2005 wrote:
You are trying to form a committee of six people at work to run a group project. If you have fifteen coworkers to choose from, how many different committees can you form?

Choosing 6 out of 15: $$15C6 = \frac{15!}{6!(15-6)!}=\frac{15!}{6!9!}=5,005$$.

21. Combinatorics/Counting Methods

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Re: A committee of three people is to be chosen from four married couples.   [#permalink] 12 May 2018, 23:47

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