[Permutations and Substitutions (other) $]$ [ Combinatorial Orbits $]$

Seventeen girls are forming a circle. In how many different ways can they stand in a circle?

#
The first method. We will fix one of the places in the circle. It is always possible to rotate the circle so that the first girl ends up in this place. The remaining 16 girls can arrange themselves in the remaining 16 places in 16! ways.

The second method. 17 girls in 17 places can be arranged in 17! ways. We will divide all these arrangements into groups, combining into one group the arrangements that can be obtained from each other by rotations. Obviously, there are 17 arrangements in each group. Therefore, the number of groups (i.e., the number of ways to stand in a circle) is \( 17! / 17 = 16! \).

## Answer

16! ways.