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A committee of 3 people is to be chosen from four married [#permalink]

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11 May 2010, 12:35

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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?

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

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 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.

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 _________________

Please give me kudos, if you like the above post. Thanks.

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

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?

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 \(B_1,B_2,C_1,C_2,D_1,D_2\).

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.

Thanks the two of you!

This is also the way I like to solve such questions and I believe it is way faster than any 10C3... and so on!

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

Hey,

can you help me how you get the 3!. What does it stand for or what does this number say?

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

Hey,

can you help me how you get the 3!. What does it stand for or what does this number say?

Thanks

It seems that you need to brush up your fundamentals:

Re: A committee of 3 people is to be chosen from four married [#permalink]

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06 Aug 2012, 11:21

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 _________________

Regards SD ----------------------------- Press Kudos if you like my post. Debrief 610-540-580-710(Long Journey): http://gmatclub.com/forum/from-600-540-580-710-finally-achieved-in-4th-attempt-142456.html

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

not sure whether this is an easy way.. but i understood this quite well except the part marked in red.

is that like if 2 men has been already selected from 2 couples we are looking for women in two ways from other two couples.

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

not sure whether this is an easy way.. but i understood this quite well except the part marked in red.

is that like if 2 men has been already selected from 2 couples we are looking for women in two ways from other two couples.

Exactly. If we choose 2 men out of 4, then the third person must be a woman from the remaining two couples: 4C2*2=6*2=12.

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

not sure whether this is an easy way.. but i understood this quite well except the part marked in red.

is that like if 2 men has been already selected from 2 couples we are looking for women in two ways from other two couples.

Exactly. If we choose 2 men out of 4, then the third person must be a woman from the remaining two couples: 4C2*2=6*2=12.

Hope it's clear.

4C2*2=6*2=12. ??? (4*3*2*1)/(2*1) = 12 but you have mentioned as 6.. is something I am missing here.

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