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

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Manager
Joined: 21 Dec 2005
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Two couples and one single person are seated at random in a [#permalink]  02 Apr 2006, 16:30
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?

Thanks
Senior Manager
Joined: 22 Nov 2005
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yes I was wrong earlier.

its 3/5

P(A) = Lets assume that all couple needs to be seated together always.

Hence single person can either seat at first place or last place.

Case 1.
When single person seats at the first place.
Number of ways in which 2 couple can seat at remaining places = 4 * 3 *2 *1.

case 2:
When single person seats at the last place
Number of ways in which 2 couple can seat at remaining places = 4 * 3 *2 *1.

Combining case 1 and case 2.

Total number of ways 2 couples can seat together = 2 * 4* 3*2 *1

Total number of ways 5 people can seat = 5* 4*3 *2 *1

P(A) = 2/5

Hence probability that 2 couple will never seat together = 3/5

Last edited by gmat_crack on 02 Apr 2006, 18:24, edited 1 time in total.
Manager
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Nope,

Manager
Joined: 20 Mar 2005
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Re: Probability [#permalink]  02 Apr 2006, 17:08
jodeci 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?

Thanks

I got 9/10
let's assume we have 5 people numbered 1 to 5 and we want to keep

(1,2) and (4,5) together therefore 3 must be first, in the middle or last

and we can order (1,2) in two ways
(4,5) in two ways

so it would be (2*2*3)/5!

equal to 1/10

so 1 - 1/10

is 9/10
Manager
Joined: 25 Mar 2006
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i got 4/15

the single person has to sit either first middle or last

case 1

first

1(because there is only 1 single person )*4(because any of the other 4 people can sit next to him ) * 2 (because out of the three only 2 that are not coupled to 4 can sit next to 4) *1( because only the person who isnt coupled to 2 can sit next to 2) and times 1 because only one person is left ) = 8

case 2
last

4*2*1*1*1 =8

case 3
and middle

4*2*1*2*1= 16

now total possible arrangements is 5! = 120

(16+8+8)/120 = 32/120 and this is = to 4/15

hope i didnt screw it up !

and 9/10 just seems a bit too close to 1 to be correct
Manager
Joined: 25 Mar 2006
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i think i know what i did wrong the single person can realy sit anywhere ...

lets say we have these people

A1 A2 B1 B2 S

it could be

S A1 B1 A2 B2

or

A1 S B1 A2 B2

and so on ... lol now i should figure our how 2 write this down
VP
Joined: 20 Sep 2005
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2/5.

Both sit together = 2*2*3! = 24.
One sit together = 2*4*3*2*1 = 48 => Since there are two couple..total nubmer of ways - 48*2 = 96.

Number of ways that at least one couple sites together = 96-24 = 72.

Number of ways that none of couple sits together = 5! - 72 = 48.

Prob. = 48/120 = 2/5.
VP
Joined: 28 Mar 2006
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The case of any couple sitting togthere = 4!*2

Since there 2 couples it is 2*4!*2

But you need to discard the case where the 2 couples are sitting together
(since we have counted that twice in the above case)

So

total -(2*Single couple sitting together) + 2 couples sitting together

120 -(2*4!*2) + 3!*2!*2!
=> 48

So probability of sitting together 48/120

So neither case = 72/120 = 3/5
Manager
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OA is 2/5 or 48/120

lhotseface

Thanks
VP
Joined: 28 Mar 2006
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oops it is 48/120....

OK the solution is like this

Let (A,B) , (C,D) and E be the people . People in () are couples

Total # of arrabgemnets = 120

Lets take the case of (AB) together So the arrgments are 4!*2!(since AB can be BA)

Same case ith (CD)

So now the case where only a single couple(either AB or CD) is together = 2*4!*2! ---------------(1)

And the case where the 2 couples are together = 3!*2!*2!-----(2)

But (2) has been counted twice in (1)
So we need to add back (2)

Finally it is 5! - (2*4!*2!) + (3!*2!*2!) = 48

Probabilty = 48/120 = 2/5
Senior Manager
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lhotseface wrote:
2/5.

Both sit together = 2*2*3! = 24.
One sit together = 2*4*3*2*1 = 48 => Since there are two couple..total nubmer of ways - 48*2 = 96.

Number of ways that at least one couple sites together = 96-24 = 72.

Number of ways that none of couple sits together = 5! - 72 = 48.

Prob. = 48/120 = 2/5.

Hi nice explanation. Can you point out where I'm making mistake.
VP
Joined: 20 Sep 2005
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At first glance, it looks like you missed the scenario that the single person can sit in the middle even when the couples sit together...

gmat_crack wrote:
lhotseface wrote:
2/5.

Both sit together = 2*2*3! = 24.
One sit together = 2*4*3*2*1 = 48 => Since there are two couple..total nubmer of ways - 48*2 = 96.

Number of ways that at least one couple sites together = 96-24 = 72.

Number of ways that none of couple sits together = 5! - 72 = 48.

Prob. = 48/120 = 2/5.

Hi nice explanation. Can you point out where I'm making mistake.
Senior Manager
Joined: 22 Nov 2005
Posts: 479
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lhotseface wrote:
At first glance, it looks like you missed the scenario that the single person can sit in the middle even when the couples sit together...

gmat_crack wrote:
lhotseface wrote:
2/5.

Both sit together = 2*2*3! = 24.
One sit together = 2*4*3*2*1 = 48 => Since there are two couple..total nubmer of ways - 48*2 = 96.

Number of ways that at least one couple sites together = 96-24 = 72.

Number of ways that none of couple sits together = 5! - 72 = 48.

Prob. = 48/120 = 2/5.

Hi nice explanation. Can you point out where I'm making mistake.

Thanks buddy. Got it.
Manager
Joined: 21 Dec 2005
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I think this method is also applicable.

Say that A.B, C.D are couples and E is single

Ways that they are adjacent to each other

A.B, C.D, E = 4*3*1*2*1 = 24

A.B, E ,C.D =4*3*1*2*1 = 24

E, A.B, C.D =1*4*3*2*1 = 24

Add them up= 24*3 = 72

Total ways to arrange 5 people in a row = 5!

5!-72= 48

Probability = 48/120 = 2/5
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