Permutations Question Thinking Question (8 students in circle) For a game of London Bridge, 8 kindergarten students form a circle holding hands and then walk in a clockwise direction. If the Prefect in charge allows the children to stand wherever they wish, in how many ways could they arrange themselves in a circle?

the cyclic permutations of $n$ distinct objects are generally $(n-1)!$

You can think of it as follows: if they were to be arranged on a line then we would have $n!$ permutations. But, the difference between a linear and a cyclic permutation is that the later have no first and last element. Starting from any one among the $n!$ linear permutations, you can create $n$ linear permutations, differing from each other only in the first and last element (consider producing them by simply moving the first element each time into the last place). These are all indistinguishable on the circle. So: $$ C_n=\frac{P_n}{n}=\frac{n!}{n}=(n-1)! $$