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# In how many ways can 6 people be arranged in a circle if 2

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Intern
Joined: 11 Jun 2007
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In how many ways can 6 people be arranged in a circle if 2 [#permalink]  14 Jun 2007, 20:05
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In how many ways can 6 people be arranged in a circle if 2 particular people are always (a) together (b) separated.
Director
Joined: 13 Mar 2007
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Schools: MIT Sloan
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Total number of ways 6 ppl can be arranged in a circle - 5! = 120

A) if a and b are always together - 4! = 24 x 2 = 48 ways

B) if they are separated - 120 -48 = 72 ways.
GMAT Club Legend
Joined: 07 Jul 2004
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Location: Singapore
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(a) Treat the two people as one entity. # of ways to arrange them = 5! = 120

(b) If two people are always seperate, then we need to rotate the other 4. So # of ways = 9*4! = 216
VP
Joined: 08 Jun 2005
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Total number of ways 6 ppl can be arranged in a circle - 5! = 120

A) if a and b are always together - 4! = 24 x 2 = 48 ways

B) if they are separated - 120 -48 = 72 ways.

when arranging items in a row , the total ways to arrange them is n!, but when arragning items in a circle the formula becomes (n-1)!.

since we have six people then (6-1)! = 120

now note that two people have to sit together - making then as one, so (5-1)! = 24

I don't understand why you are multiplying the outcome in 2. can you please explain ?

I think that the answer is 24 ways for (A).

Director
Joined: 14 Jan 2007
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KillerSquirrel wrote:
Total number of ways 6 ppl can be arranged in a circle - 5! = 120

A) if a and b are always together - 4! = 24 x 2 = 48 ways

B) if they are separated - 120 -48 = 72 ways.

when arranging items in a row , the total ways to arrange them is n!, but when arragning items in a circle the formula becomes (n-1)!.

since we have six people then (6-1)! = 120

now note that two people have to sit together - making then as one, so (5-1)! = 24

I don't understand why you are multiplying the outcome in 2. can you please explain ?

I think that the answer is 24 ways for (A).

Multiplication by 2 is required as there are two ways to seat two people together. example AB and BA.
Director
Joined: 14 Jan 2007
Posts: 779
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Kudos [?]: 76 [0], given: 0

Total number of ways 6 ppl can be arranged in a circle - 5! = 120

A) if a and b are always together - 4! = 24 x 2 = 48 ways

B) if they are separated - 120 -48 = 72 ways.

I completely agree with the solution.
Senior Manager
Joined: 21 Jun 2006
Posts: 287
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Interesting.. nice to get a formula for arrangements in circle, makes it easier. How about if the 6 people were in a queue and not a circle. What is the solution?
I believe it's 240 and 192..
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