In how many ways can 8 people be seated at a round table?



Pick one of eight people to fill a seat. Now pick one of the remaining 77 people to fill the next seat clockwise, and so on. That makes 8*7*…*2*1= 8! different arrangements in all. But there are 8 arrangements in each equivalence class, which you can get by rotating an arrangement by 0 to 7 seats. (If you rotate it by 8 seats it's the same again.)
Since there are 8! different arrangements and 8 of them form an equivalence class, there are (8-1)!=7!=5040 equivalence classes.

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