different ways can 5 beads be arranged around a necklace

[Guest]

How many different ways can 5 beads be arranged around a necklace?

I just thought itd be 4!, but when I looked back in my notes, its suppose to be 4!/2=12 - why is it divided by 2?

Thanks

 

[Denis]

Cause you can put the necklace around your neck in 2 ways: both have the same arrangement...

 

[soroban]

Hello, Anna!

This is a trick question . . .

How many different ways can 5 beads be arranged around a necklace?

I just thought itd be 4!, but when I looked back in my notes,
it's suppose to be 4!/2 = 12 --- why is it divided by 2?

It appears to be: "Arrange five people around a circular table."
. . And there are: \(\displaystyle \:4!\,=\,24\) ways.

But we're dealing with a necklace . . .

           A                     A

      E         B           B         E


        D     C               C     D

These two arrangements appear to be distinct,
. . but they are mirror-images of each other.

One of them can be obtained by turning over the other.
. . (You wouldn't have both in your jewelry box.)