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# One circular stage....there are seven positions in it, one

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CEO
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One circular stage....there are seven positions in it, one [#permalink]  14 Sep 2003, 03:34
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One circular stage....there are seven positions in it, one of them is
marked by red color and one of seven people will stand on these
positions, how many arrangements are possible?

Is it 6! / 7 ?

Senior Manager
Joined: 21 Aug 2003
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Location: Bangalore
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I know this is counter intuitive, let me first begin with a simple example of permutation in a circle.
First if it is a circle with no marking on any position answer wud be = 6!
This is because for the first person all positioins are identical/same, so he has no choice. Thus choice begins with 2nd persona who has 6 choices, 3rd 5, 4th 4 ... and so on we get total ways to arrange them = 6!

Now when one position is marked then all positions in a circle becomes unique (because of red mark). Thus from the 1st person itself we will have choices. Hence answer = 7!
something more to ponder: if clockwise and anti-clockwise does't matter then answer = 7!/2 (In 7! we assume that clockwise and anticlock wise are different arrangements)
hope this helps...
-Vicks
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Joined: 15 Aug 2003
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Kudos [?]: 719 [0], given: 781

Vicky wrote:
I know this is counter intuitive, let me first begin with a simple example of permutation in a circle.
First if it is a circle with no marking on any position answer wud be = 6!
This is because for the first person all positioins are identical/same, so he has no choice. Thus choice begins with 2nd persona who has 6 choices, 3rd 5, 4th 4 ... and so on we get total ways to arrange them = 6!

Now when one position is marked then all positions in a circle becomes unique (because of red mark). Thus from the 1st person itself we will have choices. Hence answer = 7!
something more to ponder: if clockwise and anti-clockwise does't matter then answer = 7!/2 (In 7! we assume that clockwise and anticlock wise are different arrangements)
hope this helps...
-Vicks

I am not sure ,but Akamai told us something about a rotation factor...
Dont we have the rotation factor into account when arranging things in a Circle.

Akamai..can you help!
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Location: New York NY 10024
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Re: PS : Counting Methods.. Circular Stage [#permalink]  01 Oct 2003, 02:02
praetorian123 wrote:
One circular stage....there are seven positions in it, one of them is
marked by red color and one of seven people will stand on these
positions, how many arrangements are possible?

Is it 6! / 7 ?

Vicky is correct. By marking one of the position, you eliminate the rotational symmetry of the problem, hence, the answer is simply 7!

Another way of looking at it is:

There are 7 possible people that can stand on the red spot. For each one of those, we can arrange the other 6 in 6! different ways. Hence, there are 7 x 6! = 7! ways.

Clockwise and counterwise certainly do matter -- the same way left-to-right and right-to-left would in a row. HOWEVER, if we were talking about something we could pick up and flip over, say, a bracelet or hula hoop, they we would have "mirror image" symmetry and the answer would be 7!/2.
_________________

Best,

AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

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Re: PS : Counting Methods.. Circular Stage [#permalink]  03 Oct 2003, 05:08
praetorian123 wrote:
One circular stage....there are seven positions in it, one of them is
marked by red color and one of seven people will stand on these
positions, how many arrangements are possible?

Is it 6! / 7 ?

As far as I understood the initial verbiage, ONE out of seven people will stand on the positions. ONE means 1, the only.
We have to take one out of seven and arrange him on the stage.
7*7=49, no?
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Re: PS : Counting Methods.. Circular Stage [#permalink]  03 Oct 2003, 05:36
stolyar wrote:
praetorian123 wrote:
One circular stage....there are seven positions in it, one of them is
marked by red color and one of seven people will stand on these
positions, how many arrangements are possible?

Is it 6! / 7 ?

As far as I understood the initial verbiage, ONE out of seven people will stand on the positions. ONE means 1, the only.
We have to take one out of seven and arrange him on the stage.
7*7=49, no?

There are 7 ways to pick the person to be put on the red spot and 6! ways to put the remaining 6 people. So, the answer would be 7 * 6! = 7! if direction does not matter and 7!/2 if it does.
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With the red mark, one position gets fixed where 7 persons can be arranged in 7 ways.
For rest of the 6 places,distinct items can be arranged in 6! ways. looks perfect.

[/code]
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I understand this solution, believe me. There is absolutely no need to post it twice. But this solution is OK for the following verbiage: One circular stage....there are seven positions in it, one of them is
marked by red color and seven people will stand on these
positions, how many arrangements are possible?

see a difference?
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