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At a dinner party 5 people are to be seated around a circula [#permalink]

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23 Mar 2013, 21:36

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At a dinner party 5 people are to be seated around a circular table. Two sitting arrangements are considered different only when the positions of the people are different relative to each other.What is the total number of possible sitting arrangements or the group?

You can treat this as an ordering question except that for a circular arrangement you need to divide by the number of spaces. So in this case:

5!/5=24

If you spin the circle to right, that doesn't count as a new arrangement. Dividing by the number of spaces takes that into consideration.

Happy Studies,

HG.
_________________

"It is a curious property of research activity that after the problem has been solved the solution seems obvious. This is true not only for those who have not previously been acquainted with the problem, but also for those who have worked over it for years." -Dr. Edwin Land

At a dinner party 5 people are to be seated around a circular table. Two sitting arrangements are considered different only when the positions of the people are different relative to each other.What is the total number of possible sitting arrangements or the group?

A. 5 B. 10 C. 24 D. 32 E. 120

We have a case of circular arrangement.

The number of arrangements of n distinct objects in a row is given by \(n!\). The number of arrangements of n distinct objects in a circle is given by \((n-1)!\).

From Gmat Club Math Book (combinatorics chapter): "The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle. So, for the number of circular arrangements of n objects we have:

You are very welcome. Yes, 10!/10 is exactly right for 10 people around a circular table.

HG.
_________________

"It is a curious property of research activity that after the problem has been solved the solution seems obvious. This is true not only for those who have not previously been acquainted with the problem, but also for those who have worked over it for years." -Dr. Edwin Land

At a dinner party 5 people are to be seated around a circular table. Two sitting arrangements are considered different only when the positions of the people are different relative to each other.What is the total number of possible sitting arrangements or the group?

A. 5 B. 10 C. 24 D. 32 E. 120

Check out this post on circular arrangements. It discusses why the number of arrangements is n!/n (which is the same as (n-1)!) in case there are n people sitting around a round table. http://www.veritasprep.com/blog/2011/10 ... angements/

It also discusses the relevance of this statement in the question: "Two sitting arrangements are considered different only when the positions of the people are different relative to each other"
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Re: At a dinner party 5 people are to be seated around a circula [#permalink]

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25 Mar 2013, 03:03

shelrod007 wrote:

I am a little weak in combinatorics , could some one explain to be why the answer is not 5! .

The short answer would be something like this to your question rephrased (why isn't it n! instead of n-1! ?)

The reason is that RELATIVE to each other, ie (BAC) (CAB) ie BA AB CA AC, are seated next to each other and can be considered 'one group'
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Re: At a dinner party 5 people are to be seated around a circula [#permalink]

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23 Apr 2014, 19:41

VeritasPrepKarishma wrote:

Val1986 wrote:

At a dinner party 5 people are to be seated around a circular table. Two sitting arrangements are considered different only when the positions of the people are different relative to each other.What is the total number of possible sitting arrangements or the group?

A. 5 B. 10 C. 24 D. 32 E. 120

Check out this post on circular arrangements. It discusses why the number of arrangements is n!/n (which is the same as (n-1)!) in case there are n people sitting around a round table. http://www.veritasprep.com/blog/2011/10 ... angements/

It also discusses the relevance of this statement in the question: "Two sitting arrangements are considered different only when the positions of the people are different relative to each other"

Hi Karishma,

If there were constraints such as A can't be next to B or C, does that mean that we now have 5 seats but since 3 of them are fixed, the solution would be 2!/2?

If there were constraints such as A can't be next to B or C, does that mean that we now have 5 seats but since 3 of them are fixed, the solution would be 2!/2?

I am assuming your question is this: 5 people are to be seated around a circular table such that A sits neither next to B nor next to C. How many arrangements are possible?

I don't know how you consider "...3 of them are fixed".

The way you handle this constraint would be this:

There are 5 vacant seats. Make A occupy 1 seat in 1 way (because all seats are same before anybody sits). Now we have 4 unique vacant seats (unique with respect to A) and 4 people. B and C cannot sit next to A so D and E occupy the seats right next to A on either side. This can be done in 2! ways: D A E or E A D

B and C occupy the two unique seats away from A. This can be done in 2! ways.

Total number of arrangements = 2! * 2! = 4

Check out these posts. First discusses theory of circular arrangements and next two discuss circular arrangements with various constraints:

At a dinner party 5 ppl are to seated around a circular table [#permalink]

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01 Sep 2015, 20:08

At a dinner party 5 ppl are to seated around a circular table. Two seating arrangements are considered different only when the positions of the people are different relative to each other. What is the total number of possible seating arrangements for the people.

At a dinner party 5 ppl are to seated around a circular table. Two seating arrangements are considered different only when the positions of the people are different relative to each other. What is the total number of possible seating arrangements for the people.

A.5 B.10 C.24 D.32 E.120

Merging topics. Please refer to the discussion above.

Re: At a dinner party 5 people are to be seated around a circula [#permalink]

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28 Oct 2016, 23:51

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