Circular arrangement is one of the interesting topics of Permutations. Before understanding circular arrangement let us review some important concepts of linear permutation.
For circular permutation let us consider four objects A, B, C and D. If we shift A, B, C, D one position in anticlockwise direction, then we get following arrangements.
Now, wait for a moment. Can you see that in above all four arrangements the relative position of none of the four objects A, B, C, D is changed? So, in circular arrangement they are not considered as different arrangements instead they are considered one.
Now suppose that if the above four objects A, B, C, D are to be arranged in linear, then the number of ways it can be done is 4!.
Also, in case of linear arrangements the above four circular arrangements can be shown as:
Let us generalize the above example.
Let x be the number of ways in which n different objects have to arranged in circle, taken all at a time.
For every unique circular arrangement we have n different circular arrangements in which the relative position of none of the objects is changed.
In other words, for one circular permutation there are n linear arrangements.
So, for x circular arrangements there are nx linear arrangements.
But from linear permutation n different things can be arranged in n! ways.
n! = nx
x = (n-1)!
So, if clock wise and anti clock wise arrangements are considered different, then the number of circular permutations of n different things taken all at a time is (n - 1)!.
Ring arrangement (Formation of necklace/garland using bead/flower)
Now consider the above circular arrangements, in the above discussion they were treated different because of clockwise or anti clockwise order but if no distinction is made then again the relative position of none of the objects is changed and they can be considered as same arrangement.
The above figure (i) on being flipped to the right gives figure (ii). In the above discussion figure (i) and (ii) are counted as two different arrangements which should be counted one when no distinction is allowed and hence the actual number of arrangements is (n - 1)!/2.
So, if clockwise and anti clockwise arrangement is considered same then, the number of circular permutation of n different things taken all at a time is (n-1)!/2
Now, extending the above discussion to more general:
Number of circular permutation of n different things taken r at a time
Example 1: In how many ways can 15 persons be seated around a table for dinner if there are 9 chairs?
In case of circular table the clockwise and anticlockwise arrangements will be treated as different arrangements.
Hence the total number of ways is 15P9/9
How many different necklaces can be formed from 20 beads of different colors if 10 beads are required to make a necklace?
In case of necklace there is no distinction between the clockwise and anticlockwise arrangements.
So, the required number of circular permutation is 20P10/(2 × 10)