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Can someone explain this problem to me?

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Can someone explain this problem to me? [#permalink] New post 15 Mar 2009, 18:53
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Can someone explain to me why the answer is 20 and not 5! = 120?


If 5 noble knights are to be seated at a round table, then how many different ways can they be seated?



* 120
* 96
* 60
* 35
* 24

Fix a knight in a spot to arrange the rest. The formula is (N-1)! , or (5-1)! = 4! = 24 .
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Re: Can someone explain this problem to me? [#permalink] New post 16 Mar 2009, 08:45
Lets take five letters A,B, C,D and E as 5 knights, these letters can be arranged as ABCDE or EABCD, which are two different arrangements on a single line.
If these are written along the periphery of a circle, the two arrangements are one and same.Figure below:

Image

These type of permutations are called as CIRCULAR PERMUTATIONS and are different only when the relative order of the objects is changed.

Hence, the position of one object is fixed and the remaining (n-1) objects are arranged in all possible ways and the number of distinct permutations is (n-1)!
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Re: Can someone explain this problem to me? [#permalink] New post 16 Mar 2009, 09:24
benbo1234 wrote:
Can someone explain to me why the answer is 20 and not 5! = 120?


If 5 noble knights are to be seated at a round table, then how many different ways can they be seated?



* 120
* 96
* 60
* 35
* 24

Fix a knight in a spot to arrange the rest. The formula is (N-1)! , or (5-1)! = 4! = 24 .


The question is poorly written. Depending on the interpretation, the answer could be N! or (N-1)!. If you're looking for the number of possible arrangements among specific chairs (e.g., if the chairs were labelled), then N!. If you're looking for # of possible arrangement of the knights relative to each other (e.g., if the chairs weren't labeled, i.e., were indistinguishable), then (N-1)!. I would have chosen N!, because maybe their table is next to the king's and perhaps they're fighting over who faces the king!
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Re: Can someone explain this problem to me? [#permalink] New post 20 Nov 2010, 18:45
I'm with OhThatMBA on this one. The question is poorly written. Even though the table is a circle, the seats are not equal. Each arrangement can be rotated clockwise one position and that would be a different seating. Therefore 5! would be my answer.
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Re: Can someone explain this problem to me? [#permalink] New post 20 Nov 2010, 22:50
hi benbo!

circular permutation is a type of permutation which has no starting point and no ending point. It is a set of elements that has an order, but no reference point. It circles back around on itself and encloses.

i see it as a pretty straight q, so the solution will be (n-1)! given that u seat one of the knights and set him as a reference point ,it is 4! which is 24.

pls orrect me if i am wrong.
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Re: Can someone explain this problem to me? [#permalink] New post 21 Nov 2010, 20:06
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In case nothing is mentioned, in a circular arrangement, two seating arrangements are considered different only when the positions of the people are different relative to each other.
That is, in the question above, you will use the formula (n - 1)!

Here is the theory behind it:

Arranging 3 people (A, B, C) in a row: Following 6 ways
A B C, A C B, B A C, B C A, C A B, C B A
3! ways

Why is arranging 3 people in a circle different?
Look at one arrangement of 3 people A, B and C around a small round table O. Ignore the dots.
....A
....O
B......C
If I am B, A is to my left, C is to my right.

Look at this one now:
....C
....O
A......B
Here also, if I am B, A is to my left and C is to my right.
In a circle, these are considered a single arrangement because relative to each other, people are still sitting in the same position. This is the general rule in circular arrangement.
You use the formula n!/n = (n - 1)! because every n arrangements are considered a single arrangement. e.g. if n = 3, the given 3 arrangements are the same:
.....A ................ C ............... B
.....O ................ O .............. O
B........C ........ A ..... B ..... C........ A

In each of these, if I am B, I am sitting in the same position relative to others. A is to my left and C is to my right.
and these three are the same:
.....C ................ A ............... B
.....O ................ O .............. O
B........A ........ C ..... B ..... A........ C

Here, if I am B, C is to my left and A is to my right. Different from the first three.
Hence no. of arrangements = 3!/3 = 2 only

You might need to use n! in a circle if they mention that each seat in the circular arrangement is numbered and is hence different etc. Then there are just n distinct seats and n people. If nothing of the sorts is mentioned, you always use the (n - 1)! formula for circular arrangement.
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Re: Can someone explain this problem to me? [#permalink] New post 22 Nov 2010, 10:51
(N-1)! therefore, (5-1)! = 4! => 24 ways..
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Re: Can someone explain this problem to me? [#permalink] New post 22 Nov 2010, 17:12
Thank you for the explanation VeritasPrepKarishma. I do understand what you're saying but my issue is with the wording of the question: "how many ways can the knights be seated" is not as clear as "what is the # of possible arrangements of the knights relative to each other". I believe that a proper gmat question should be crystal clear so that even a person who has never laid eyes on the exam would know what the question is asking. As OhThatMBA said, "perhaps they're fighting over who faces the king" or over who faces the window or whatever!
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Re: Can someone explain this problem to me? [#permalink] New post 22 Nov 2010, 19:02
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suesie970 wrote:
Thank you for the explanation VeritasPrepKarishma. I do understand what you're saying but my issue is with the wording of the question: "how many ways can the knights be seated" is not as clear as "what is the # of possible arrangements of the knights relative to each other". I believe that a proper gmat question should be crystal clear so that even a person who has never laid eyes on the exam would know what the question is asking. As OhThatMBA said, "perhaps they're fighting over who faces the king" or over who faces the window or whatever!


Yes suesie970, I absolutely agree. If the questions clearly states "The seats are numbered" or "different arrangements relative to each other", great! If it doesn't, don't worry. An accepted connotation of 'circular arrangement' is 'arrangement relative to each other'.
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Re: Can someone explain this problem to me?   [#permalink] 22 Nov 2010, 19:02
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