Prove that there are $(n-1)!$ possible cyclic orders of $n$ objects

Question

I'd like to prove that there are $(n-1)!$ possible cyclic orders of $n$ objects.

I started with a proof by induction, but I am not sure if this is the right way to approach this problem.

I showed that this held for the $n=2$ case, because if we have two objects there is only one way to rearrange them. Then I assumed that it held for the $n=k$ case, i.e. that there were $(k-1)!$ different cyclic orderings. Now I need to prove that it holds for the $n=k+1$ case.

Answer 1

There are $n!$ ways to place $n$ distinguishable objects in a linear sequence. Now "glue" the ends of the sequence to form a cycle. You can rotate any sequence into $n$ equivalent positions.

$n!/n = (n-1)!$.

Answer 2

Well, without loss of generality, one can put, say, the element $n$ to the first position. Then there are $(n-1)!$ cyclic orders of $n$ elements.