Circular Permutations

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When things are arranged in places along a line with first and last place, they form a linear permutation. So far we have dealt only with linear permutations. When things are arranged in places along a closed curve or a circle, in which any place may be regarded as the first or last place, they form a circular permutation.

The permutation in a row or along a line has a beginning and an end, but there is nothing like beginning or end or first and last in a circular permutation. In circular permutations, we consider one of the objects as fixed and the remaining objects are arranged as in linear permutation.

Thus, the number of permutations of 4 objects in a row = 4!, where as the number of circular permutations of 4 objects is (4-1)! = 3!.

The following arrangements of 4 objects O₁, O₂, O₃, O₄ in a circle will be considered as one or same arrangement.

Observe carefully that when arranged in a row, O₁ O₂ O₃ O₄, O₂O₃O₄ O₁, O₃O₄O₁O₂, O₄O₁O₂O₃ are different permutations. When arranged in a circle, these 4 permutations are considered as one permutation.

Theorem:

The number of circular permutations of n different objects is (n-1)!.

Proof:

Each circular permutation corresponds to n linear permutations depending on where we start. Since there are exactly n! linear permutations, there are exactly  permutations. Hence, the number of circular permutations is the same as      (n-1)!.

Example 1:

1. Three boys and three girls are to be seated around a table in a circle. Among them, the boy X does not want any girl as neighbour and girl Y does not want any boy as neighbour. How many such arrangements are possible?

Suggested answer:

The arrangement is as shown in the figure, the boy X will have B₂, B₃ as neighbours. The girl Y will have G₂, G₃ as neighbours. The two boys B₂, B₃ can be arranged in two ways. The two girls G₂, G₃ can be arranged in two ways.

Hence, the total number of arrangements = 2 x 2 = 4.

Example 2:

In how many different arrangements can 6 gentlemen and 6 ladies sit around a table if

i) there is no restriction and

ii) no two ladies sit side by side?

Suggested answer:

i) Here, the total number = 6 + 6 = 12.

12 persons can be arranged in circular permutation as (12 - 1)! = 11! ways.

ii) When 6 gentlemen are arranged around a table, there are 6 positions, each being between two gentlemen for 6 ladies, when no two ladies sit side by side. Now, the number of ways in which 6 gentlemen can be seated around a table = (6 - 1)! = 5!.

Then, corresponding to each seating arrangement for the gentlemen, the 6 ladies can be seated in 6! ways.

The required number of arrangements = (5!)(6!)

= (120)(720) = 86400