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by mmslf75 » Sat Dec 26, 2009 5:49 am
At a dinner party 5 people are to be seated arround a circular table. Tow seating arrangemets are considered different only when the positions of the people are different relative to each other. What is the total number of different possible seating arrangements for the group?

5,10,24,32,120

Answer is 24

Query 1

When we have people seated around in a circular arrangement should we consider it as

( n - 1 ) ! ??

therefore, here , we have 4 * 3 * 2 = 24 ways right ??

Query 2

Ways of arranging people around a circular table and People in a linear manner are different ??
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by Stuart@KaplanGMAT » Sat Dec 26, 2009 7:22 pm
mmslf75 wrote:At a dinner party 5 people are to be seated arround a circular table. Tow seating arrangemets are considered different only when the positions of the people are different relative to each other. What is the total number of different possible seating arrangements for the group?

5,10,24,32,120

Answer is 24

Query 1

When we have people seated around in a circular arrangement should we consider it as

( n - 1 ) ! ??

therefore, here , we have 4 * 3 * 2 = 24 ways right ??
correct!
Query 2

Ways of arranging people around a circular table and People in a linear manner are different ??
It's different because of the way circular arrangements work.

Let's call our chairs 1, 2, 3, 4 and 5 and our people A, B, C, D, E.

One arrangement is:

1A 2B 3C 4D 5E

another arrangement is:

1B 2C 3D 4E 5A

if we just used n! to count the number of possible arrangements, these would be counted as two different permutations. However, if we look at how the people are seated relative to each other, these are actually the exact same arrangement.

So, for each of the n! arrangements, there are n duplicates.

So, for circular arrangements, the formula plays out as:

n!/n = n*(n-1)!/n = (n-1)!
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by mmslf75 » Mon Dec 28, 2009 7:33 am
Stuart Kovinsky wrote:
mmslf75 wrote:At a dinner party 5 people are to be seated arround a circular table. Tow seating arrangemets are considered different only when the positions of the people are different relative to each other. What is the total number of different possible seating arrangements for the group?

5,10,24,32,120

Answer is 24

Query 1

When we have people seated around in a circular arrangement should we consider it as

( n - 1 ) ! ??

therefore, here , we have 4 * 3 * 2 = 24 ways right ??
correct!
Query 2

Ways of arranging people around a circular table and People in a linear manner are different ??
It's different because of the way circular arrangements work.

Let's call our chairs 1, 2, 3, 4 and 5 and our people A, B, C, D, E.

One arrangement is:

1A 2B 3C 4D 5E

another arrangement is:

1B 2C 3D 4E 5A

if we just used n! to count the number of possible arrangements, these would be counted as two different permutations. However, if we look at how the people are seated relative to each other, these are actually the exact same arrangement.

So, for each of the n! arrangements, there are n duplicates.

So, for circular arrangements, the formula plays out as:

n!/n = n*(n-1)!/n = (n-1)!
hey.. got it..
thanks stuart!!
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by funx » Mon Dec 28, 2009 7:39 pm
Could someone please explain the second sentence? i.e. seating arragements are considered different only when the positions of the people are different relative to each other.

Without even getting into the math, which is actually very simple as I noticed, I was bogged down by the phrasing.
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