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.

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Karishma

Veritas Prep | GMAT Instructor

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