Question

How many different necklaces can we make from n beads of different colors? Consider two necklaces the same if (like in a circular arrangement) one can be obtained from the other via rotation or if (unlike in a circular arrangement) one can be obtained from the other via flipping the necklace over.

Answer 1

`((n -1)!)/(2)`

Explanation:

Let us consider n beads of different colors, arranged in a line. If the beads were to be arranged in a straight line, there are n! ways to do this.

Now, if the beads were to be in a circular arrangement, a pattern will repeat n times.

So the number of different circular arrangements can be obtained by dividing by n, such that we get
`(n!)/(n)` =
`(n-1)!`

The patterns can be obtained via rotating either clockwise or anticlockwise, therefore 2 ways. So we can divide the total by 2.

Hence, the different number of necklaces which we can make from n beads of different colors is
`((n -1)!)/(2)`