It we are to line-up N people then it is easy to see there are 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 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 people left.

Starting at the right-hand side of the head-seat, there are ways to seat the remaining people.