Circular Permutation

Let’s say you own three blue, black, and red pens, as well as two pencils. One of the pencils belongs to firm A, while the other belongs to company B. Your friend then invites you to select one of the products. How many various ways can you do it? Because you can choose any one of the given items at a time, the answer, in this case, is five. Your friend then wants you to put them in a box. How many different ways can you do it? Permutations are a challenge that involves arranging things in a certain way. This section will discuss permutations. 

Permutations

A permutation is essentially an arrangement of objects in a specific order in which only a few of them are chosen at once. We count the number of ways an arrangement could happen in a permutation.

Consider the pens and pencils in the previous scenario. How will we know how many different ways these items could arrange together in a box if we need to know? There are two options: choose a pencil first, choose a pen first, and place them wherever we like.

Circular Permutation

The setups mentioned above are linear. Some arrangements have a circular nature to them. Observe the roundtable discussion about crafting a necklace out of several coloured beads. It is similar to putting things in a closed loop. A circular permutation is the number of ways connected with a circular arrangement. 

Let’s say there are four chairs all around the roundtable, but we need to seat four people: A, B, C, and D. Any of the jobs can all be filled by anyone. However, as soon as one takes a seat, the other three people’s selections are reduced by one. Finally, there is only one chair available for the last individual.

Arrangement in Circular Permutation

If four people A, B, C, and D are organised in a row in the above situation, they can do so in four different ways. There’s also a beginning and an end in a row arrangement, just as in a linear permutation. We must examine the perspectives of all those involved in the arrangement. However, there is no such thing as a beginning or an endpoint in the circular permutation.

We assume one person or thing to be fixed and the remaining people arranged in the circular permutation. Assume that A’s position is fixed. The number of arrangements the other three people can make (4 − 1)! = 3! = 6 when one of them has a fixed position.

So the four individuals A, B, C, and D can organise themselves in six different ways. This holds even if we set the positions of B, C, or D.

The clockwise and anticlockwise arrangement determines the person’s positioning in the preceding example. If there is no such dependency, the number of arrangements drops to   ½ (n-1)! =½ (4-1)! = 3/2 = 3. This will be the case if the person’s position is unaffected by the arrangement’s order. It’s similar to the way beads of the same colour are strung together in a necklace.

Conclusion

A circular permutation is the number of ways to place n distinct objects around a fixed circle. There are two types of it. Case 1: The order of clockwise but also anticlockwise rotations differs. Case 2: Orders are the same in both clockwise and anticlockwise directions. However, there is no such thing as a beginning or an end in the circular permutation. We assume one person or something to be fixed and the remaining people arranged in the circular permutation. The number of different arrangements the other three people can make When one of them is in a fixed position, the result is (4 1)! = 3!