If I have 10 different colored beads. In how many ways can I make a necklace?
The number of circular permutations of n objects is n! /2 if the clockwise and anti clockwise arrangement are the same. Now in the case of beads in a lace whether the necklace is worn clockwise or turned over and worn in the other way the result would be the same. If you’re wondering where that would not be the same is a group of people seated in a circular formation.
Therefore the number of ways in which the beads can be arranged is 10! / 2 = 1814400
This the permuations around a ring.
Unlike the arrangements in a line the arrangements around a ring , the starting and end points coincide.
Therefore the order of the place does not count.
Now, let there be n different (coloured) beads.
So for a fixed place of the one particular bead in the necklace, the number of arrangements of other n-1 beads = (n-1)! ways.
If consider clockwise and anticlock wise as of no consequence, then n different beaded necklace = could be in (n-1)!/2.
Therefore 10 different coloured beaded necklace could be in
(10-1)! = 9! ways = 362880 ways (cw and antcw are considered different)
9!/2 = 181440 ways (cw and anti cw are not consedred different.)