Maths > Permutations and Combinations > 4.0 Circular Permutations
Permutations and Combinations
1.0 Factorial notation
2.0 Basic Principle of Counting
3.0 Permutation
4.0 Circular Permutations
5.0 Combinations
6.0 Restricted Selection
7.0 Restricted Arrangement
8.0 Greatest Term
9.0 Possible selections from $n$ distinct objects
10.0 Possible Selection from $n$ Identical Objects
11.0 Possible Selection from $n$ Objects having Distinct and Identical Objects
12.0 Total Number of Possible Divisors for a Given Natural Number
13.0 Sum of all Possible Divisors of a Natural Number
14.0 Exponent of a Prime Number in $n!$
15.0 Division and Distribution of Objects
16.0 Multinomial Theorem
4.1 Permutation when taken all at a time
The linear arrangements have a beginning and ending. In circular permutations, it is not so.
Consider ${a_1},{a_2},{a_3}.......,{a_{n - 1}},{a_n}$.
This has $n!$ permutations for linear arrangement.
In circular arrangement, $\{ {a_1},{a_2},{a_3}.......,{a_{n - 1}},{a_n}\} ,\{ {a_2},{a_3},{a_4}.......,{a_n},{a_1}\} ,.....,\{ {a_n},{a_1},{a_2}.......,{a_{n - 2}},{a_{n - 1}}\} $ , are one and the same.
If total number of permutations without uniqueness is $n!$, and $y$ is unique permutations. Then,
$$(n)(y) = n!$$
Since each unique permutation repeats $n$ times.
Thus total unique circular permutations, $$ = \frac{{n!}}{n} = (n - 1)!$$
Difference between clockwise and anti-clockwise arrangements
Consider the following arrangements:
When there is distinction in object arranged clockwise and anticlockwise then the number of permutations is, $$ = (n - 1)!$$
In case there is no distinction in clockwise and anti-clockwise arrangement then the number of permutations is, $$ = \frac{{(n - 1)!}}{2}$$
Note: To find whether there is any difference in clockwise and anti-clockwise direction, check whether the arrangement can be viewed from only one side or from either sides.
For example: Arranging people in circular table can be viewed only from one side, whereas beads in a necklace can be viewed from both the sides.
Table arrangement given below can be viewed only from one side.
Here first and second necklaces are the same arranged viewed from either sides and the third is totally a different arrangement.
Note: When positions are numbered the circular arrangement is considered linear.
