Circular permutation

Jun 16, 2020, 16:45 IST

Circular permutation

Definition :-The arrangements we have considered so far are linear. There are also arrangements in closed loops, called circular arrangements.
Consider four persons A, B C and D, who are to be arranged along a circle. It’s one circular arrangement is as shown in adjoining figure.

Formula description :-
Circular permutation is the total number of ways in which n distinct objects can be arranged around a fix circle. It is of two types.

1. Case (a): - Clockwise and Anticlockwise orders are different.
2. Case (b): - Clockwise and Anticlockwise orders are same.

Case (a): formula :-
Pn = (n – 1)!
Where –
• Pn = represents circular permutation
• n = Number of objects
There are

Case (b): formula :-
Where –
• Pn = represents circular permutation
• n = Number of objects

Example 1 :-In how many ways can 6 people be seated at a round table?

Solution :-
the number of ways will be (6 – 1)!, or 120.

Example 2 :-20 persons we invited to a party. In how many ways can they be seated in a round table such that two particular persons sit on either side of the host?

Solution :-
After fixing the places of three persons (1 host + 2 persons) and treating them as 1 unit we can arrange the total (20 - 2 + 1) = 19 units in 18! ways. Again these particular persons can sit on either side of the host in 2 ways.
Hence the total number of ways is 18! × 2.