Solved Example of Circular Permutation

Example 1: How many different ways are there to arrange 8 guys around a circular table?

Solution:

Eight guys sit around a circular table = (8-1)! = 7!

⇒ Eight guys sit around a circular table = 7 x 6 x 5 x 4 x 3 x 2 x 1

⇒ Eight guys sit around a circular table = 5040 ways

Example 2: There should be seven valuable stones. Make sure that every stone is a diamond. Find out how many different configurations there are for these diamonds.

Solution:

Here, all the diamonds are the same in the predicament. This implies that it is impossible to tell whether the stones are arranged in a clockwise or anticlockwise fashion. Thus, in this case, we use the second calculation to determine how many different ways the stones may be stacked.

Here n = 7

Consequently, the formula provides the number of potential circular permutations.

P_n = (n – 1)! / 2!

⇒ P_n = (7 – 1)! / 2!

⇒ P_n = 6! / 2!

⇒ P_n = 360

Example 3: Calculate the circular permutation of 6 people seated around a round table while (i) If the anticlockwise and clockwise orders are different. (ii) If the anticlockwise and clockwise orders are the same.

Solution:

Case 1: If the anticlockwise and clockwise orders are different. Here n = 6. Use the Formula

P_n = (n – 1)!

⇒ P₆ = (6 – 1)!

⇒ P₆ = (5)!

⇒ P₆ = 5 x 4 x 3 x 2 x 1

⇒ P₆ = 120

Case 2: If the anticlockwise and clockwise orders are the same. Here n = 6. Use the Formula

P_n = (n – 1)! / 2!

⇒ P₆ = (6 – 1)! / 2!

⇒ P₆ = 5! / 2!

⇒ P₆ = 60

Circular Permutation is an arrangement notion in which the objects are arranged in a closed loop. The beginning and end points are ambiguous, in contrast to linear layouts. Since one item can be fixed and the others positioned around it, there are (n-1)! circular permutations of the ‘n’ objects. 5 persons seated around a circle-shaped table, for instance, produces 4! or 24 variants. It is important to distinguish between circular and linear permutations since the counting method is impacted by the arrangement’s cyclical nature. This idea offers a unique viewpoint on how to organize parts in a circular layout and has applications in a variety of industries, including seating arrangements, clock setups, and molecular structures.