Circular Permutations


So far, we’ve only considered arrangements in a line or a row. This lesson will talk about arrangement of (distinct) objects in a circle.

Circular Permutations

Consider the problem: In how many ways can 5 persons be seated at a round table? Or, in how many ways can 5 distinct objects be arranged in a circle?

The arrangements differ only in the relative order of the objects, and the relative spacing between the objects does not matter.

To answer the question, consider a fictitious tire company called ANODE, which has to print its company name on its tires.

The five linear arrangements – ANODE, NODEA, ODEAN, DEANO and EANOD will be identical when arranged in a circular fashion. Have a look at the figure below.

pc6-circular-permutations

Each of the tires can be rotated to align with any one of the rest. That means all of these arrangements are identical, even though we started with different linear arrangements.

Therefore 5 linear arrangements will correspond to only one circular arrangement.

Coming back to the question – In how many ways can 5 distinct objects be arranged in a circle?

Now there are 5! or 120 different linear arrangements possible. Therefore the number of circular arrangements will be 5! ÷ 5. (As every 5 linear arrangements will correspond to 1 circular arrangement). This is same as 4! or 24.

To generalize, the number of arrangements of n distinct objects in a circle will be n! / n, or (n – 1)!

Lesson Summary

The number of arrangements of n distinct objects in a circle will be n! / n, or (n – 1)!

I’ll cover a few examples related to circular arrangements in the next lesson. See you there.


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