Permutation and Combination - Circular Permutation

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Circular Permutations

When (say) ‘n’ items are arranged in a circle, the number of arrangements of these ‘n’ items compared to arranging them in a straight line will be lesser. The reason for this is that in a straight line we can have two different arrangements namely ABCD and DABC, whereas in a circle these two arrangements will be considered the same. One good example to understand this concept better will be to consider a necklace which is studded with four different coloured diamonds in its periphery. Let us assume that the different colours are red, blue, black and green. Now, let us consider two possibilities.

Possibility 1: The necklace is spread on a table as a straight line. Now, we know that the total number of possible arrangements will be 4! = 24 arrangements.

Possibility 2: You are wearing the necklace around your neck with a seamless transition between any two diamonds. Now, we can clearly see that the number of arrangements reduces since you will not be able to make out the difference between two different straight line arrangements namely “red, blue, black, green” and “blue, black, green, red” in the necklace you are wearing around your neck.

Now, how do we get the number arrangements in the second possibility described above? For that, we need to fix the position of one diamond say green. Now, the remaining 3 diamonds can be arranged in 3! = 6 ways. Now, even if we rotate the necklace in either direction, the arrangement we will get will be different because one item will be fixed and will not move.

To generalize, we can say that the number of ways in which ‘n’ distinct items can be arranged in a circle will be (n-1)!. Now, this has been obtained assuming that there is a difference between clockwise and anti-clockwise arrangements. Suppose we assume that the necklace is made up of distinct but round pearls. Now, there will be no difference between clockwise and anti-clockwise arrangements because when I have a clockwise arrangement and turn it around, I will get an anti-clockwise arrangement and I cannot say the difference between these two as the pearls are round (symmetrical about all axes). Hence, the total number of arrangements of ‘n’ items in a circular fashion with no difference between clockwise and anti-clockwise arrangements will be half the number of arrangements in the previous case and hence will be (n-1)!/2.