At a dinner party, 5 people are to be seated around a circular table.

C

let 1,2,3,4,5 are people.

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.

Therefore, N=12*2=24

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

Therefore, N=12*2=24

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.

I was confused by the wording of the question "only when the positions of the people are different relative to each other".
I knew the formula (n-1)!, but I though that the correct answer would require some limited set of the combinations, defined by "only when the positions of the people are different relative to each other".

Then if the question did not mentioned this special condition of ""only when the positions of the people are different relative to each other", the answer would be 24x5=120. Right?

"the positions of the people are different relative to each other" just means different arrangements (around a circular table). The number of arrangements of n distinct objects in a circle is \((n-1)!=4!=24\), (120 would be the answer if they were arranged in a row).

Arranging 3 people (A, B, C) in a row:
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?

....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, 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

Here, they have mentioned 'relative to people' only to make it clearer. In a circle, anyway only relative to people arrangements are considered.
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.

Hi there,

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.

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?

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

Please provide feedback to see if this makes sense:

I approached it thinking, if you have a circle, say, ABCDE there would be 5! ways of arranging, however, the question states that an arrangement is only different if the positions relative to each other are different.

So (1/5)th of the time each person in the circle would in the same position relative to another person. Therefore I did (1/5)x5! = 24

I was thinking "symmetry" as well - what are the experts thoughts on this? The formula though is definitely the easier way.

Hi icetray,

Your thinking on this question is just fine. Conceptually, since we're dealing with a circular table with 5 chairs (and not a row of chairs), the table could have 5 different "starting chairs." As such the arrangements (going around the table):

ABCDE
BCDEA
CDEAB
DEABC
EABCD

Are all the same arrangement (just 'revolved' around the table). Since we're NOT allowed to count each of those (they're not different arrangements, they're just rotations of the same arrangement), we have to divide the permutation by 5.

5!/5 = 24

This type of 'set-up' is relatively rare on Test Day - there's a pretty good chance that you won't see it at all. If you do though, then your way of handling the "math" is just as viable as the formula that was given.

GMAT assassins aren't born, they're made,
Rich

When determining the number way to arrange a group around a circle, we subtract 1 from the total and set it to a factorial. Thus, the total number of possible sitting arrangements for 5 people around a circular table is 4! = 24.

Answer: C

Solution:

This question is simply about arranging 5 people across a circular table.

n people can be arranged across a table in (n-1)! ways

=>5 people can be arranged in (5-1)! = 4! =24 ways

(option c)

Devmitra Sen
GMAT SME

Although we can quickly apply the circular arrangement formula (i.e., number of ways to arrange n objects in a circle = (n - 1)!), we can also solve the question using the Fundamental Counting Principle (FPC, aka the slot method). In the process of doing so, you'll also learn WHY the circular arrangement formula works

First label the five chairs as follows:

We can seat the first guest in one of the 5 available chairs.
We can seat the next guest in one of the 4 remaining chairs.
We can seat the next guest in one of the 3 remaining chairs.
We can seat the next guest in one of the 2 remaining chairs.
We can seat the last guest in the 1 remaining chair.
So, the total number of ways to seat the guests = (5)(4)(3)(2)(1) = 120 ways

The answer, however, is NOT E, because we have inadvertently counted every possible arrangement 5 times.

For example, the five arrangements shown here...

... are all the same, because the relative positions of the five people are the same in each case.

Since we have counted each unique arrangement 5 times, we must divide 120 by 5 to get 24 possible arrangements

Answer: C

To solve this problem, we need to consider the concept of circular permutations.

The formula for the number of circular permutations of n objects is:
(n-1)!

In this case, we have 5 people, so the number of possible seating arrangements is:
(5-1)! = 4! = 24

Therefore, the correct answer is C. 24.