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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.

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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 ADCBBACD BADC BCAD BCDA BDAC BDCACABD CADB CBAD CBDA CDAB CDBADABC DACB DBAC DBCA DCAB DCBAIf 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 waysNote 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.

Notice that the first group in row 1 (highlighted in yellow) corresponds to the circle on the left in the first pair, and the first group in row four (also highlighted in yellow) corresponds to the circle on the right in the first pair. There is a group in yellow in the second and third rows as well that correspond to the second pair of circles.

This means that there are three duplicates of each grouping from row one. Rotating the circle does not create new arrangements. If we divide the total by 4, this will result in the number of distinct (unique) arrangements in a circle.

4! / 4 = 3! = (3) * (2) * (1) = 6Now you can see all of the arrangements from row one:

The Formula

We now have the formula. The permutation of n items in a row is n!, but the permutation of n items in a circle is n!/n or (n-1)!.

This will be true whether the items are letters, numbers, colors, objects, people, etc.

The Circle Is Free

This circle is considered to be fixed. This is the most common situation, and this is what is intended unless otherwise specified. It cannot be lifted out of the plane and flipped over.

If it could, it would mean that the circle is free and that both the clockwise and counterclockwise arrangements would result in duplicates. In our diagram of the six circles, the first and sixth circles would be the same, the second and fourth would be the same, and the third and fifth would be the same if the circle is free. The resulting number of arrangements is three, or half of the amount from before.

Therefore, the permutation of n items in a ‘free’ circle (one that can be flipped over) is n! / 2n or (1/2) * (n-1)!

Example Problem

How many ways can five different colored chairs be arranged in a fixed circle? In a free circle?There are five chairs, which means that n = 5.Permutations of 5 items in a fixed circle is (5-1)! = 4! = (4) * (3) * (2) * (1) = 24 waysPermutations of 5 items in a free circle is (5-1)! / 2 = 4! / 2 = 24 / 2 = 12 ways

Lesson Summary

The number of ways a group of items can be arranged depends on several factors. The formula for each scenario is different.

  • Does the order matter? Permutations of n items in a line = n!
  • Are the items in a line or in a circle? Permutations of n items in a fixed circle = n! / n = (n – 1)!
  • Is the circle free? Permutations of n items in a free circle = n! / 2n = (1 / 2) * (n-1)!

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