Circular Permutation of n Different Things Taken r at a Time

Permutations and combinations are fundamental concepts in combinatorics, which is a branch of mathematics that deals with counting, arrangement, and probability. When we arrange objects in a circle, the permutations are referred to as circular permutations. Circular permutations differ from linear permutations because the position of items is relative to each other rather than absolute.

Understanding Circular Permutations

In a circular permutation, we consider one arrangement to be the same as another if you can rotate the first arrangement to get the second one. This means that the starting point is not fixed as it is in linear permutations.

Linear vs. Circular Permutations

Aspect	Linear Permutation	Circular Permutation
Order	Absolute position matters	Relative position matters
Fixed Point	Starting point is fixed	No fixed starting point
Rotation	Different arrangements	Same arrangement
Formula	$P(n, r) = \frac{n!}{(n-r)!}$	$P(n) = (n-1)!$ for n things

Formulas for Circular Permutations

When dealing with circular permutations, the formulas differ based on whether the objects are distinct or identical, and whether the direction (clockwise or counterclockwise) is considered.

1. Distinct Objects (Direction Matters): $$ P(n) = (n-1)! $$ This formula is used when there are n distinct objects, and the direction of arrangement matters (i.e., clockwise and counterclockwise arrangements are considered different).

2. Distinct Objects (Direction Does Not Matter): $$ P(n) = \frac{(n-1)!}{2} $$ This formula is used when there are n distinct objects, but the direction of arrangement does not matter (i.e., clockwise and counterclockwise arrangements are considered the same).

3. r Objects Taken from n Distinct Objects (Direction Matters): $$ P(n, r) = \frac{n!}{(n-r)!} $$ This is a general permutation formula where r objects are taken from n distinct objects, and the direction of arrangement matters.

4. r Objects Taken from n Distinct Objects (Direction Does Not Matter): $$ P(n, r) = \frac{n!}{2(n-r)!} $$ This formula is used when r objects are taken from n distinct objects, and the direction of arrangement does not matter.

Examples

1. Example of Distinct Objects (Direction Matters): Suppose we have 5 distinct beads to be arranged on a bracelet (which is circular). The number of ways to arrange these beads is given by: $$ P(5) = (5-1)! = 4! = 24 $$

2. Example of Distinct Objects (Direction Does Not Matter): If we consider the same 5 beads on a necklace (where flipping the necklace gives the same arrangement), the number of ways to arrange these beads is: $$ P(5) = \frac{(5-1)!}{2} = \frac{4!}{2} = 12 $$

3. Example of r Objects Taken from n Distinct Objects (Direction Matters): If we have 6 distinct flowers and we want to arrange 4 of them in a circular garland, the number of ways to do this is: $$ P(6, 4) = \frac{6!}{(6-4)!} = \frac{6!}{2!} = 360 $$

4. Example of r Objects Taken from n Distinct Objects (Direction Does Not Matter): Using the same 6 flowers, if we want to arrange 4 of them in a circular garland where the direction does not matter, the number of ways is: $$ P(6, 4) = \frac{6!}{2(6-4)!} = \frac{6!}{2 \cdot 2!} = 180 $$

Important Points to Remember

• In circular permutations, the reference point is not fixed, and rotation does not create a new arrangement.
• When considering direction, arrangements that can be rotated into each other are counted separately if they look different when rotated (e.g., clockwise vs. counterclockwise).
• When direction does not matter, arrangements that can be rotated into each other or are mirror images are considered the same.
• The number of circular permutations of n different things taken all at a time is (n-1)!. If the direction does not matter, it is (n-1)!/2.
• When taking r objects from n distinct objects, the general permutation formula is used, adjusted for circular arrangements.

Understanding circular permutations is essential for solving problems related to round tables, necklaces, rings, and other circular arrangements where the relative position of objects is the primary concern.