Circular permutations
There are two cases of circular-permutations:-
(a) If clockwise and anti clock-wise orders are different, then total number of circular-permutations is given by (n-1)!
(b) If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by (n-1)!/2!
Proof(a):
(a) Let?s consider that 4 persons A,B,C, and D are sitting around a round table
Shifting A, B, C, D, one position in anticlock-wise direction, we get the following agreements:-
Thus, we use that if 4 persons are sitting at a round table, then they can be shifted four times, but these four arrangements will be the same, because the sequence of A, B, C, D, is same. But if A, B, C, D, are sitting in a row, and they are shifted, then the four linear-arrangement will be different.
Hence if we have ?4? things, then for each circular-arrangement number of linear-arrangements =4
Similarly, if we have ?n? things, then for each circular ? agreement, number of linear ? arrangement = n.
Let the total circular arrangement = p
Total number of linear?arrangements = n.p
Total number of linear?arrangements
= n. (number of circular-arrangements)
Or Number of circular-arrangements = 1 (number of linear arrangements)
n = 1( n!)/n
circular permutation = (n-1)!
Proof (b) When clock-wise and anti-clock wise arrangements are not different, then observation can be made from both sides, and this will be the same. Here two permutations will be counted as one. So total permutations will be half, hence in this case.
Circular?permutations = (n-1)!/2
Note: Number of circular-permutations of ?n? different things taken ?r? at a time:-
(a) If clock-wise and anti-clockwise orders are taken as different, then total number of circular-permutations = nPr /r
(b) If clock-wise and anti-clockwise orders are taken as not different, then total number of circular ? permutation = nPr/2r
Moderation Team
![[Post New]](/templates/default/images/icon_minipost_new.gif){kind=icon}
![[Up]](/templates/default/images/icon_up.gif "[Up]"){kind=icon}
90
1
