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Re: In how many ways to choose a group of 3 people from 6 couples so that [#permalink]
10 Aug 2006, 18:28

Let me try....

6 couples = 12 people.

There are three slots to be filled.

If we start with any of the 12, the second person will have to be out of 10 because the spouse of the first one chosen must be left out. Same philosophy for the third person brings it to 8.

Re: In how many ways to choose a group of 3 people from 6 couples so that [#permalink]
11 Aug 2006, 06:17

OK, I think the fallacy in my logic above is that I assumed it was a perrmutation rather then a combination. In my approach, XYZ is different from ZXY, even though you have the same set of 3 with no couples.

Since there were 6 couples to mitigate the permuations, (12*10*8)/6 = 160.

Re: In how many ways to choose a group of 3 people from 6 couples so that [#permalink]
11 Aug 2006, 12:45

Professor's approach is easier:

Men Women
2 (6 men and 2 to choose = 6C2) 1 (4C1)
This one woman being selected cannot be the wife of either of the two men selected earlier. Therefore that leaves you with 4 women to choose from (discarding 2 women who are wives of the two men selected)
= Total 6C2 * 4C1 = 60

1 (4C1)similar logic to above 2 (6C2)
= Total 6C2 * 4C1 = 60

Re: In how many ways to choose a group of 3 people from 6 couples so that [#permalink]
17 Mar 2015, 10:08

Just stumbled across this from a google search, so I'll add my technique.

first step: you can only choose one person from each couple, so I found how many combinations of couples I can have:

6c3=20

So there are 20 different ways to combine 3 of the 6 couples.

second step: once I have chosen the 3 couples to combine I have to choose man/wife from each. There are 2 choices in each of the 3 couples so I think of it as a binary number where 0=female and 1=male:

Re: In how many ways to choose a group of 3 people from 6 couples so that [#permalink]
17 Mar 2015, 20:48

2

This post received KUDOS

Expert's post

mand-y wrote:

In how many ways to choose a group of 3 people from 6 couples so that no couple is chosen

Another single step calculation method is that you can select them using the simple basic counting principle (which arranges them in 1st person, 2nd person and 3rd person) and then you can divide by 3! to un-arrange.

First person can be selected in 12 ways.

Second person in 10 ways (since the first person selected and his/her partner are not available)

Third person in 8 ways (since the first and second people and their partners are not available)

Total number of ways = 12*10*8/3! = 160 _________________

Re: In how many ways to choose a group of 3 people from 6 couples so that [#permalink]
14 Jul 2015, 02:06

VeritasPrepKarishma wrote:

mand-y wrote:

In how many ways to choose a group of 3 people from 6 couples so that no couple is chosen

Another single step calculation method is that you can select them using the simple basic counting principle (which arranges them in 1st person, 2nd person and 3rd person) and then you can divide by 3! to un-arrange.

First person can be selected in 12 ways.

Second person in 10 ways (since the first person selected and his/her partner are not available)

Third person in 8 ways (since the first and second people and their partners are not available)

Total number of ways = 12*10*8/3! = 160

Why do you divide it by 3!? what do you mean by un-arrange?

Re: In how many ways to choose a group of 3 people from 6 couples so that [#permalink]
14 Jul 2015, 02:47

1

This post received KUDOS

Expert's post

patufet6 wrote:

In how many ways to choose a group of 3 people from 6 couples so that no couple is chosen

Another single step calculation method is that you can select them using the simple basic counting principle (which arranges them in 1st person, 2nd person and 3rd person) and then you can divide by 3! to un-arrange.

First person can be selected in 12 ways.

Second person in 10 ways (since the first person selected and his/her partner are not available)

Third person in 8 ways (since the first and second people and their partners are not available)

Total number of ways = 12*10*8/3! = 160

Why do you divide it by 3!? what do you mean by un-arrange?

Thanks

When we use the method of placement of objects like 12*10*8 then we MUST take the cognizance of the arrangement of the objects already included in the method. e.g. 5*4*3 = 5C3*3! e.g. 10*9*8 = 10C3*3!

Similarly, 12*10*8 = Selected of three individuals such that no two form couple INCLUDING Arrangement of those three Individuals

therefore, we have to exclude the arrangements of those three individuals (3!) because the question only demand the no. of possible groups and not the arrangement of group members as well.

So only selection of those three individuals so that they are not couple = 12*10*8 / 3! = 160

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