Why do we subtract 1 when calculating permutations in a ring? 
$10$ persons are to be arranged in a ring shape. Number of ways to do that is $9!.$

I wonder why we subtarct $1$ in all such cases.
I can imagine that if A,B,C,D are sitting in a row then B,C,D,A would give me a different combination but had they been sitting on a circular table then both the above combinations would imply the same thing. Same for CDAB or DABC.
But how does it all lead to $(n-1)!$ formula? Individual cases, I am able to imagine but overall how are we generalizing it?

Imagine that $4$ persons are siting in a round table, you can order them in 4! ways, but each time you rotate the table one person to the right, it's the same ordering.
How many rotations are there? as the number of persons, $4$. So the total number of ways is $\frac{4!}{4}=3!$. 
The same way for 10 persons, and also $n$ persons as the general case.
