Permutations and Combination

Hi Roland2rule,

We're told that 5 people are to be seated around a circular table and that two sitting arrangements are considered different only when the positions of the people are different relative to each other. We're asked for the total number of possible sitting arrangements of the group.

If the 5 people were sitting in a straight-line, then we'd be dealing with a standard Permutation question - and there would be 5! = 120 possible arrangements. Here, we're dealing with a circular table, so any of the 5 chairs could be the "first chair" and 1 arrangement of people could be created in 5 different ways. Thus, we have to divide 120 by 5 to determine the number of unique arrangements. 120/5 = 24 arrangements.

Final Answer: C

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

BTGmoderatorRO wrote:At a dinner party, 5 people are to be seated around a circular table. two seating arrangement are considered different only when the position of the people are different relative to each other. what is the total number of possible seating arrangement for the group?

a.5
b.10
c.24
d.32
e.120

When determining the number of ways 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 seating arrangements for 5 people around a circular table is (5 - 1)! = 4! = 24.

Answer: C

BTGmoderatorRO wrote: ↑

Sun Dec 17, 2017 5:22 am

At a dinner party, 5 people are to be seated around a circular table. two seating arrangement are considered different only when the position of the people are different relative to each other. what is the total number of possible seating arrangement for the group?

a.5
b.10
c.24
d.32
e.120

OA is A
Why is A correct. Can I use possibly Venn's diagram to analyze this question?Pls, I want an Expert to reply me on this.

ASIDE: In its long history, the GMAT has made public only 1 circular arrangement question, and you're looking at it.

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