Circular Permutation - When Object Are Different :
Permutation of objects in a row is called as linear permutation. If we arrange the objects along a closed curve it is called as circular permutation.
Thus in, circular permutation, we consider one object fixed and the remaining objects are arranged as in the case of a linear arrangements.
`text(Theorem-1 :)`The number of circular permutation of n distinct objects is `(n -1) !`
Proof :-
Consider `5` objects `A, B, C, D, E` to be arranged around a closed curve is called circular permutation.
Let the total number of circular permutation be `x`. Above circular permutation is equivalent to `5` linear permutations given by `ABCDEF, EABCD, DEABC, CDEAB, BCDEA`
that is one circular permutation is equivalent to `5x` linear permutation given by
`x . 5=5!`
`x=(5!)/5=(5 . (5-1)!)/5=(5-1)!`
Similarly forn objects `nx = n !`
`x=(n!)/n=(n-1)!`
(i) `n` distinct things taken all at a time and arranged along circle in `(n - 1) !` ways
(il) Taken `r` things out of `n` distinct things at a time and arranged along circle in `text()^nC_r (r - 1)!` ways.
`text(Theorem-2 :)`
If anticlockwise and clockwise are considered to be same total number of circular permutation given by
`((n-1)!)/2`
If we arrange flowers or garland beads in a neckless then there is no distinction between clockwise anticlockwise direction.
Note:-
(i) If we haven diffrent things taken rat a time in form of a garland or neckless
Required number of arrrangements `=(text()^nC_r . (r-1)!)/2`
(il) The distinction between clockwise and anticlockwise is ignored when a number of people have to be seated around a table so as not to have the same neighours.
Thus in, circular permutation, we consider one object fixed and the remaining objects are arranged as in the case of a linear arrangements.
`text(Theorem-1 :)`The number of circular permutation of n distinct objects is `(n -1) !`
Proof :-
Consider `5` objects `A, B, C, D, E` to be arranged around a closed curve is called circular permutation.
Let the total number of circular permutation be `x`. Above circular permutation is equivalent to `5` linear permutations given by `ABCDEF, EABCD, DEABC, CDEAB, BCDEA`
that is one circular permutation is equivalent to `5x` linear permutation given by
`x . 5=5!`
`x=(5!)/5=(5 . (5-1)!)/5=(5-1)!`
Similarly forn objects `nx = n !`
`x=(n!)/n=(n-1)!`
(i) `n` distinct things taken all at a time and arranged along circle in `(n - 1) !` ways
(il) Taken `r` things out of `n` distinct things at a time and arranged along circle in `text()^nC_r (r - 1)!` ways.
`text(Theorem-2 :)`
If anticlockwise and clockwise are considered to be same total number of circular permutation given by
`((n-1)!)/2`
If we arrange flowers or garland beads in a neckless then there is no distinction between clockwise anticlockwise direction.
Note:-
(i) If we haven diffrent things taken rat a time in form of a garland or neckless
Required number of arrrangements `=(text()^nC_r . (r-1)!)/2`
(il) The distinction between clockwise and anticlockwise is ignored when a number of people have to be seated around a table so as not to have the same neighours.