How many ways are there to arrange n distinct objects in a CIRCLE?

We recall that the number of ways to order n distinct objects in a STRING is n! (n factorial, n.(n-1).(n-2)...2.1 Familiarity with this is prerequisites for tackling this question)We begin, as is best in any situation, with a simple example, before we look at the general form.Say we have four distinct objects, ABCD We have 4! = 24 such ways of arranging these objects distinctly into strings, so no two are equivalent.eg, A-B-C-D =/= B-C-D-AHowever, now lets turn each of these strings into CIRCLES.eg the string A-B-C-D becomes -A-B-C-D- where you can imagine that the "*" represents where D is connected to A, thus giving a circle.You can now see that if we apply this to our other string in the example, B-C-D-A, it becomes -B-C-D-A- where * is as above.After writing this on paper, it should be clear that in fact these circles are exactly the same (just different by rotation).If we continue this process to all 24 strings of ABCD we see that they fall into 6 groups, with each group containing 4 equivalent circles.So the number of ways to arrange 4 distinct objects into circles.Now how about the general case?In our example, the number of circles of 4 distinct objects = (the number of strings of 4 distinct objects)/4This is true in the general case too. To understand why, note the following. Say we have a circle of length n. How many distinct strings can be made from this circle? Well, to turn the circle into a string, we would have to cut it at some point. There are n objects, so we have n places we could cut. This means that (the number of strings of n distinct objects) = (the number of circles of n distinct objects) x nFor our final answer, we recall that the number of strings of n distinct objects = n!So the number of circles of n distinct objects = n!/n which, as we can cancel n in the top and bottom, gives us (n-1)!EXTENSION PROBLEM- What is we considered two circles to be the same if we considered their mirror image equivalent? How many distinct, UN-isomorphic rings would we have?