IV. The numbers of permutations of N distinct objects arranged in a circle is (n-1)!
The Question: Why is it that in a circle the formula to be use is (n-1)! why -1?
Spoiler:
It we are to line-up N people then it is easy to see there are $\displaystyle N!$ ways to do it.
But sitting N people around a table there is no starting (left-hand most) person.
If we wanted to we could designate a ‘starting’ person in N ways.
Thus there are only $\displaystyle \frac{N!}{N}=(N-1)!$ different ways to do this.
Here is another way to look at this problem.
At a circular table with N seats, pick one to be the head of the table.
Select one person to be seated there.
Now there are $\displaystyle N-1$ people left.
Starting at the right-hand side of the head-seat, there are $\displaystyle (N-1)!$ ways to seat the remaining people.