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John draws a regular five pointed star in the sand, and at each of the 5 outward-pointing points and 5 inward-pointing points he places one of ten different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent?

 Jun 22, 2018
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AoPS Solution

 

There are \(10!\) ways to put the shells in the sand, not considering rotations and reflections. Arrangements can be reflected or not reflected and can be rotated by 0, 1/5, 2/5, 3/5, or 4/5, so they come in groups of ten equivalent arrangements. Correcting for the symmetries, we find that there are \(\frac{10!}{10}\)\(=\boxed{3628800}\) distinct arrangements.

 Jun 22, 2018

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