For those acquainted with Permutations and Combinations,

. . but unfamiliar with the "Circular Table Problem",

. . this is a classic "trick question."

There are six people to be seated in six chairs set around a circular table.

In how many ways can they be seated?

We must understand that a *rotation* of an arrangement is __not__ a different arrangement.

. . These two seating are considered equivalent. Code:

A B F A
* - - * * - - *
/ \ / \
F* *C E* *B
\ / \ /
* - - * * - - *
E D D C

The reasoning is like this . . .

The first person can sit anywhere . . . it doesn't matter.

. . Then the other five can be seated in $\displaystyle 5! = 120$ ways.

With a bracelet, an additional "trick" is involved.

Once again, rotations are not counted.

In addition, *reflections* are also not counted.

. . These two bracelets are considered equivalent. Code:

A B B A
* - - * * - - *
/ \ / \
F* *C C * *F
\ / \ /
* - - * * - - *
E D D E

With twelve (distinguishable) beads on a bracelet,

. . there are: .$\displaystyle 11!$ circular arrangements.

And we must eliminate reflections.

. . There are: .$\displaystyle \frac{11!}{2} \:=\:19,958,\!400$ possible bracelets.