Question

It is known that n different objects can be arranged in a circle in (n-1)! ways. How many ways can (n + 1) different objects be arranged?

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Answer 1

The number of ways to arrange (n+1) different objects in a circle is (n+1)!.

This can be seen by considering the following two cases:

• The (n+1)th object is placed at the beginning of the circle. In this case, there are (n)! ways to arrange the remaining n objects.
• The (n+1)th object is placed somewhere in the middle of the circle. In this case, there are (n)! ways to arrange the first n objects, and (n)! ways to arrange the last n objects. However, since the (n+1)th object can be placed in any of n positions, we need to multiply this by n to get (n^2)!.

Adding these two cases together, we get (n+1)! ways to arrange (n+1) objects in a circle.