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The applications of Burnside's formula in counting orbits has wide applications (I believe). But, whatever the books I followed on Group Theory, many (or almost all) of the applications mentioned in them are for "coloring problem" which involves roughly coloring vertices of a regular $n$-gon with different colors.

Q. What are the other simple applications of Burnsides theorem which I can present to undergraduates while teaching group theory?

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If you allow for Polya's enumeration theorem also here are a list of interesting problems that have been answered:

How many ways can you put 8 red, 6 green and 7 blue balls in 4 indistinguishable bins? (A sweet solution by Marko Riedel)

The classical problem of necklaces and braceletets . Also an explanation.

Using it to count graphs (An incredible solution by Marko Riedel)

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