How Many Unique 5 White and 5 Black Bead Necklaces Can Be Made?

[jameshaley]

Homework Statement

How many necklace with 5 white beads and 5 black beads can be constructed?

Homework Equations

Circular Permutation problem

The Attempt at a Solution

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I did 10!/5!5!=252

but from there I didn't get anywhere.

I know this includes repeats from rotational symmetry and reflections. but i am not really sure how to get rid of these.

i try dividing by 10 but it gives 25.2, which does not make sense to me.

 

[haruspex]

There is no simple way to analyse this sort of problem. You need to break it down into cases according to the symmetries.
You know there must somewhere be a black and a white adjacent, so you could fix on such a pair. That gets you down to 8-choose-4 immediately. Then it's a matter of removing duplicates.

 

[haruspex]

Thinking some more about this... consider how many symmetries any given pattern might have. If only one (i.e. the identity) how many times will the given unique pattern be counted in your 252? What if two symmetries in the group? Etc. Then it's a matter of figuring how many patterns have each of the symmetry counts.
Fwiw, I make the final answer 16.