## Circular Permutation

**Lesson no. 4: Circular Permutations:**

How can *A, B, C* be arranged around a circle?

Not in the three ways as shown above! Why?

Because each one of *A, B, C* has the same neighbor!

Without changing neighbor, only changing seats will not change the circular permutation.

Change neighbors and you will change the circular permutation. As follows:

So, three persons *A, B, C* can only be arranged in 2 ways around a circle.

*In a formula type it is:*

Generalize. How? As follows?

*n different things can be arranged around a circle in (n – 1)! ways.*

In other words, with n different things, there are:

**Clock wise and Anti-Clock wise permutations:**

Consider the figure:

You thought the circular permutations change on changing neighbors.

Not really. Why?

Changing neighbors can change circular permutations only when you treat clock wise direction different from the anti-clock wise.

What if you treat the two directions same?

Then only one circular permutation!

Generalize. How? Like below:

*Without treating the two directions differently, n different things can be arranged around a circle in many*

ways.

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