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

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In how many ways can 6 people be arranged in a circle if 2 [#permalink] New post 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.
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 [#permalink] New post 14 Jun 2007, 20:26
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.
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 [#permalink] New post 14 Jun 2007, 22:02
(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
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 [#permalink] New post 14 Jun 2007, 22:18
grad_mba 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.


hi grad_mba

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).

:-D
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 [#permalink] New post 15 Jun 2007, 02:20
KillerSquirrel wrote:
grad_mba 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.


hi grad_mba

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).

:-D


Multiplication by 2 is required as there are two ways to seat two people together. example AB and BA.
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 [#permalink] New post 15 Jun 2007, 02:21
grad_mba 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.


I completely agree with the solution.
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 [#permalink] New post 18 Jun 2007, 08:14
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..
  [#permalink] 18 Jun 2007, 08:14
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