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Say you have N white balls and N black balls. How many distinguishable ways can I arrange them in a circle? I consider 2 arrangements to be distinguishable if no rotation or reflection can make them the same.

If I were to arrange them in a line, I think there are $(2N)!/4$.

For N=2, there are 2 ways to arrange them: BWBW and WWBB.

For N=3, I think there are 4 ways: WWWBBB, WBWBWB, WWBBWB, WWBWBB

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  • $\begingroup$ Hint: for any unique solution on a line, wrap it onto a circle. No any rotation preserves that circle solution. How many different line solutions map to the same circle solution? $\endgroup$
    – kjetil b halvorsen ♦
    Commented May 2, 2020 at 4:46
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    $\begingroup$ What constitutes a "distinguishable" way? Would two configurations that differ by a rotation be distinguishable (the group $C_n$)? How about two configurations that differ by a reflection (the group $D_n$)? BTW, there is a very general, powerful, and effective method to answer questions of this sort: Polya's Enumeration Theorem. $\endgroup$
    – whuber ♦
    Commented May 2, 2020 at 15:44
  • $\begingroup$ I added a clarification about what distinguishable means. I forgot about reflections. Thanks for the link to PET! $\endgroup$
    – user3433489
    Commented May 2, 2020 at 19:03

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After following some links from the Polya Enumeration Theorem, I came across this wikipedia page on necklaces and bracelets, which are well studied in combinatorics:

https://en.wikipedia.org/wiki/Necklace_(combinatorics)

Apparently I was describing a bracelet.

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