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

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26 Dec 2007, 07:16

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At a dinner party, 5 people are to be seated around a circular table. 2 seating arrangements 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?

At a dinner party, 5 people are to be seated around a circular table. 2 seating arrangements 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?

If the people sit in a row instead of a circular table, we would have 5*4*3*2*1 = 120 different combinations. But (restriction), if the people spin around the table without being in different position relative to each other does not generates a new position. Then, we have to eliminate ABCDE, BCDEA, CDEAB, DEABC and EABCD. Total: 5 arrangements. Then, instead of 5*4*3*2*1 = 120, we have to eliminate the 5 arrangements --> 5*4*3*2*1 = 24

After reading the solution given by walker, I thought in this one:

Fixing 1 person in the table (C for example), and moving the neighbours of this person.

Slots: 12345 --> We fix the third slot with person "C" --> 12C34--> then we have four possibilities for the slot 1 (person A, B, D or E), three possibilities for the second slot (ABDE less the person that we placed in the slot 1), two for the fourth slot and one for the last slot:

4*3*1*2*1 = 24

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:

OA is C, good...anyway, provided I am quite bad at combs, would you please explain it to me step by step in a very clear way..i cannot catch your 3 points!

let 1,2,3,4,5 are people.
"_ _ _ _ _" - positions

1. we fix position of 1
"_ _ 1 _ _"

2. we have 4*3=12 possible positions for left and right neighbors of 1.
"_ x 1 _ _" x e {2,3,4,5}. 4 variants "_ x 1 y _" y e {(2,3,4,5} - {x}. 3 variants total number of variants is 4*3=12

3. for each position of x1y we have 2 possible positions for last two people: ax1yb and bx1ya.
or "a x 1 y _" a e {(2,3,4,5} - {x,y}. 2 variants "a x 1 y b" b e {(2,3,4,5} - {x,y,a}. 1 variants

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

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08 Jul 2013, 06:24

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At a dinner party, 5 people are to be seated around a circular table. 2 seating arrangements 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?

If the people sit in a row instead of a circular table, we would have 5*4*3*2*1 = 120 different combinations. But (restriction), if the people spin around the table without being in different position relative to each other does not generates a new position. Then, we have to eliminate ABCDE, BCDEA, CDEAB, DEABC and EABCD. Total: 5 arrangements. Then, instead of 5*4*3*2*1 = 120, we have to eliminate the 5 arrangements --> 5*4*3*2*1 = 24

After reading the solution given by walker, I thought in this one:

Fixing 1 person in the table (C for example), and moving the neighbours of this person.

Slots: 12345 --> We fix the third slot with person "C" --> 12C34--> then we have four possibilities for the slot 1 (person A, B, D or E), three possibilities for the second slot (ABDE less the person that we placed in the slot 1), two for the fourth slot and one for the last slot:

4*3*1*2*1 = 24
_________________

Encourage cooperation! If this post was very useful, kudos are welcome "It is our attitude at the beginning of a difficult task which, more than anything else, will affect It's successful outcome" William James

At a dinner party, 5 people are to be seated around a circular table. 2 seating arrangements 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?

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

Soln: Since the arrangement is circular and 2 seating arrangements are considered different only when the positions of the people are different relative to each other, we can find the total number of possible seating arrangements, by fixing one person's position and arranging the others.

Thus if one person's position is fixed, the others can be arranged in 4! ways. Ans is C.

1. we fix position of 1
2. we have 4*3=12 possible positions for left and right neighbors of 1.
3. for each position of x1y we have 2 possible positions for last two people: ax1yb and bx1ya.

At first lets see, there are how many cases to sit, if the they will sit not in round table. there are 5*4*3*2*1=5! cases And because it is no important that which sit is first that is mean that we must devise this to 5, prob=5!/5=4!=24

The solution Araik mentioned is logical as this was a problem with a round table. As a rule one needs to understand that there is a difference is arrangement in a row and that of a circle. Please refer to Math Workbook for 'COMBINATORICS'. Its explained very well there.

sort cut for any circular seating arrangements: (n-1)! =(5-1)! = 4! = 4*3*2*1 = 24

hi, kinda skeptical about the shortcut...do you really mean that when we see any circular table question, we can just plug in the (n-1)! formula?

So if it is a 8 table round table sitting arrangement, I can just use (8-1)! to get question resolved? Any other criteria or...does it limit only to certain criteria? Pls enlighten...many thanks!

sort cut for any circular seating arrangements: (n-1)! =(5-1)! = 4! = 4*3*2*1 = 24

hi, kinda skeptical about the shortcut...do you really mean that when we see any circular table question, we can just plug in the (n-1)! formula?

So if it is a 8 table round table sitting arrangement, I can just use (8-1)! to get question resolved? Any other criteria or...does it limit only to certain criteria? Pls enlighten...many thanks!

(n-1)! is correct for this constraint free question irrespective of the number of people in question. However, you may have to apply some logic if there are other constraints, such as A can't sit with B, OR D and E must sit together.
_________________

OA is C, good...anyway, provided I am quite bad at combs, would you please explain it to me step by step in a very clear way..i cannot catch your 3 points!

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