People around a round table

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(a) In how many ways can $8$ people be seated around a round table, if all arrangements that differ only by a rotation are considered the same?

(b) In how many ways can $8$ all-different beads be strung on a circular bracelet, if besides rotations we also consider flips equivalent (the bracelet can be turned over)?

Solution

(a) Fixing one person as a reference to remove equivalent rotations, the other $7$ must be arranged in a line: $N_{\text{tavolo}} = (8-1)! = 7! = \boxed{5040}.$

(b) On the bracelet each arrangement and its mirror image coincide, so we divide further by $2$: $N_{\text{braccialetto}} = \frac{7!}{2} = \frac{5040}{2} = \boxed{2520}.$