Why is the arrangement of $n$ things round a circular table is $(n - 1)!$ & not $n!$ ? Suppose there are $n$ seats round a table . We have to find the number of circular permutations of $n$ different men taken all at a time when clockwise & anti-clockwise orders are different.
Since this is a circular table, we have to first fix the position of one man taking the position of which relative positions of other men can be found. We can choose the 1st man in $n$ ways. Subsequently others can be chosen in $(n - 1)!$ ways . Therefore total ways $$ n\cdot(n - 1)! = n!$$ . 
But my theory didn't work as my book wrote that the total number of circular arrangements is $(n - 1)!$ ways. Why is it so? What is the problem with my theory? How did the book write $(n - 1)!$ ?? Please explain me.

But each permutation is taken as being the same as the other $n-1$ permutations which arise when you rotate it, so you have to divide by $n$ - only relative positions matter.
Alternatively, given any permutation, you can count it clockwise from person $1$ to fit in the other $n-1$ people.
