How many ways can six friends be arranged around a circular?

Vincen wrote:How many ways can six friends be arranged around a circular dinner table?

A. 16
B. 48
C. 96
D. 120
E. 720

The OA is D.

I have a huge doubt. Why D is not the right option? Is it different because of the circular table?

I think you wish to say why is E (720) not the correct answer?

An arrangement is a row is different than that on a circular table.

If 6 friends are to be seated in a row, the number of arrangements = 6! = 720.

However, if the same number of people, 6, are to be seated on a dining table, the number of ways = (6 – 1)! = 5! = 120.

The number of ways = (6 – 1)! because, unlike with an arrangement in a row, mere shifting one by place by all the friends does not get you a new arrangement.

Let's understand this way.

Say the 6 friends A, B, C, D, E, and F are seated on a 6-seat bench this way: A B C D E F.

If A, B, C, D, and E moves one position to their right, F will have to take A's position, so you get a new arrangement: F A B C D E.

However, if these 6 friends A, B, C, D, E, and F are seated at a round table this way: A B C D E F, and each moves to its left, the new arrangement would still be A B C D E F-- no new arrangement!

The correct answer: D

Hope this helps!

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Vincen wrote: ↑

Fri Sep 08, 2017 2:36 pm

How many ways can six friends be arranged around a circular dinner table?

A. 16
B. 48
C. 96
D. 120
E. 720

The OA is D.

I have a huge doubt. Why D is not the right option? Is it different because of the circular table?

Recall that the circular permutations formula for n people is (n - 1)!.

Since the friends are to be seated around a circular table, the number of ways that they can be arranged is (6 - 1)! = 5! = 120 ways.

Answer: D