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# Two couples and one single person are seated at random in a

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Director
Joined: 05 Jan 2005
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Two couples and one single person are seated at random in a [#permalink]  04 Mar 2005, 15:11
Two couples and one single person are seated at random in a row of five chairs. What is the probability that neither of the couples sits together in adjacent chairs?

I know sample space is 120 (5!) and hence the denominator. What do we manipulate to get the numerator, given the info. above?
Director
Joined: 19 Nov 2004
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Location: SF Bay Area, USA
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I think this question is from Princeton CAT.
If so, kindly post which test you took this from in the heading so that people who haven't taken the test can avoid it.
VP
Joined: 30 Sep 2004
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there are many ways to calculate this. i would do it like this:

all outcomes= 5!

outcomes where the single person is in first + last position: 4c1*2c1*1c1*1c1*2=16 => it┬┤s the same for both positions because it is symmetric

outcomes where the single person is in second and fourth position: 4c1*2c1*1c1*1c1*2=16 => => it┬┤s the same for both positions because it is symmetric

outcomes where the single person is in third position: 4c1*2c1*2c1*1c1=16

48/120
Director
Joined: 05 Jan 2005
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Oh please pardon me guys. I didn't remember to include the source. It's from PR CAT 1.
Senior Manager
Joined: 25 Oct 2004
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is it 96/120

Whats the OA?
Director
Joined: 05 Jan 2005
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OA is 2/5. Sorry for the delay.
SVP
Joined: 03 Jan 2005
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Re: Prob. [#permalink]  09 Mar 2005, 09:19
Arsene_Wenger wrote:
Two couples and one single person are seated at random in a row of five chairs. What is the probability that neither of the couples sits together in adjacent chairs?

I know sample space is 120 (5!) and hence the denominator. What do we manipulate to get the numerator, given the info. above?

Outcomes one couple is sitting together: 4!*2
Outcomes the other couple is sitting together: 4!*2
Outcomes that both couples are sitting together: 3!*2*2

Outcomes neither couples are sitting together
=5!-4!*2-4!*2+3!*2*2
=5!-(24-6)*4=120-72=48

Probability =48/120=2/5
Director
Joined: 21 Sep 2004
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Hongu can u please explain how is this
Outcomes that both couples are sitting together: 3!*2*2
SVP
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Since both of the couples must sit together with each other, you can arrange the two husband and the single person, for example, and then put the wifes beside their husband. The arrangement would be P(3,3). There are two ways to put the first wife beside the first husband, and two ways to put the second wife beside the second husband, thus *2*2.
Manager
Joined: 11 Jan 2005
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Honghu is the man.

Thanks dude.

Manager
Joined: 24 Jan 2005
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Location: Boston
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The woman you mean!!!
:)
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