If I have 10 different colored beads. In how many ways can I make a necklace?

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.)