Circular Permutation: Formula & Examples

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  • 0:00 What Is a Permutation?
  • 1:13 What Is the Effect of…
  • 2:55 The Formula
  • 3:17 The Circle Is Free
  • 4:04 Example Problem
  • 4:35 Lesson Summary
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Lesson Transcript
Instructor: Eric Istre

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

This lesson introduces the concept of circular permutations along with its formula. Learn how to determine the number of ways to arrange items in a circle, and when you are through take the quiz to see what you have learned.

What Is a Permutation?

How many ways can the letters A, B, C, and D be arranged in a line? If the order of the letters is changed, does that make a new arrangement? Since the letters are all different, changing the order does make a new arrangement. Thus, we call this type of question a permutation because the order of the items matters. A permutation is just an arrangement of n items in which the order matters.

Here are all of the possible ways to arrange these four letters:

ABCD ABDC ACBD ACDB ADBC ADCB

BACD BADC BCAD BCDA BDAC BDCA

CABD CADB CBAD CBDA CDAB CDBA

DABC DACB DBAC DBCA DCAB DCBA

If you count up the number of different arrangements, you will get 24. Another way to reach this conclusion is to think of it like this. There are four letter choices for the first position in the group, three letter choices left for the second position, two letter choices left for the third position, and finally, one letter left for the fourth position. This allows us to create an equation to get our answer:

(4) * (3) * (2) * (1) = 4! = 24 ways

Note that 4! means 4 factorial, which is just another way to write (4) * (3) * (2) * (1).

What Is the Effect of Putting the Items in a Circle?

What if the letters were arranged in a circle instead? Would this affect the number of different possible arrangements? This type of situation is called a circular permutation, which is simply finding the arrangement of things in a circle. Look at these two circle arrangements. At first glance, they appear to be different, but ask yourself another question. Is it possible to rotate the figure on the left to make it look like the figure on the right? Since the answer is yes, this means that the letters are in the same position relative to each other; therefore, this is not a new arrangement.

As a matter of fact, there are four arrangements that are the same. Here are the other two.

In this diagram, each arrangement (from the original 24) that is the same in a circle is also highlighted in the same color so you can see the duplicates more easily.

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