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Dinner Table- Combinations

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by DevB » Sun Jul 27, 2014 11:44 am
At a dinner party, 5 people are to be 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 different possible seating arrangements for the group?

1. 5
2. 10
3. 24
4. 32
5. 120

Qs is from GMAT Prep. Pls help in answering.
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by Brent@GMATPrepNow » Sun Jul 27, 2014 11:57 am
DevB wrote:At a dinner party, 5 people are to be 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 different possible seating arrangements for the group?

A 5
B 10
C 24
D 32
E 120
The number of arrangements of n distinct objects in a circle = (n - 1)!

So, for this question, the total number of different possible seating arrangements of 5 people = (5 - 1)! = 4! = 24

Answer: C
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by [email protected] » Sun Jul 27, 2014 1:11 pm
Hi DevB,

The "math" behind Permutation questions will be influenced by the restrictions that the prompt gives you to work with, so you have to pay careful attention to how these questions are worded.

Here, we know that there are 5 people and that they're sitting around a circular table. This is different from 5 chairs that are in a row (which would be a much easier calculation).

With 5 seats in a row, we'd have 5x4x3x2x1 = 120 arrangements....BUT since we have a circular table, some of these arrangements are duplicates. With 5 chairs, there are 5 "starting spots" that would lead to duplicates. For example...

ABCDE
BCDEA
CDEAB
DEABC
EABCD

These 5 options are actually the SAME option, since the letters are sill in order "going around" the table. We can't count this as 5 options; it's just 1 option.

So the math becomes: 5!/5 = 5x4x3x2x1/5 = 24

Final Answer: C

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by phanikpk » Wed Jul 30, 2014 6:22 am
In a circular arrangement,

No. of possible combinations include (n-1)!

So, (5-1)!= 4!= 4*3*2*1= 24
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by GMATinsight » Wed Jul 30, 2014 7:30 am
DevB wrote:At a dinner party, 5 people are to be 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 different possible seating arrangements for the group?

1. 5
2. 10
3. 24
4. 32
5. 120

Qs is from GMAT Prep. Pls help in answering.
CONCEPT:
The linear arrangements of n Objects are described by n x (n-1) x (n-2)...3 x 2 x 1 = n!
The reason for that is first object has n places and then the next object has (n-1) ways and so on

But In Circular arrangements, if all the objects to arranged remain movable then the order will repeat (e.g.if all objects shift their position by one place around circle in clockwise direction). Therefore we have to fix the position of one object out of all objects (n) to be arranged around circle.

Therefore, only (n-1) objects remain movable
hence, The arrangements of n objects around circle is ideally arrangement of (n-1) objects = (n-1)!

i.e. Arrangement of 5 people around Circle = (5-1)! = 4! = 4.3.2.1 = 24

Answer: Option C
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by GMATGuruNY » Thu Jul 31, 2014 2:56 am
DevB wrote:At a dinner party, 5 people are to be 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 different possible seating arrangements for the group?

1. 5
2. 10
3. 24
4. 32
5. 120
For circular permutations, we generally count arrangements RELATIVE TO THE FIRST PERSON SEATED.
Step 1: Place person A at the table.
Step 2: Count the number of ways to arrange the REMAINING people.

In the problem above, after person A is seated, the number of ways to arrange the remaining 4 people = 4! = 24.

The correct answer is C.

Another problem:
At a dinner party, Amy, Bob, Cindy, David, Ellen, and Frank are to be 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 different possible seating arrangements for the group if Amy must sit directly across from Bob?
After Amy is seated:
Number of options for Bob = 1. (He must sit directly across from Amy.)
Number of ways to arrange the remaining 4 people = 4! = 24.
To combine these options, we multiply:
1*24 = 24.

Another:
At a dinner party, 3 men and 3 women are to be 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 different possible seating arrangements for the group if no man may sit adjacent to another man?
After man A is seated:
Number of women who could be seated to the right of man A = 3.
Number of remaining women who could be seated to the left of man A = 2.
Number of remaining women who could be seated directly across from man A = 1.
Number of ways to arrange the remaining 2 men = 2! = 2.
To combine these options, we multiply:
3*2*1*2 = 12.
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