Circular permutation with numbered seats Suppose we have $n$ people sitting on a round table? What is the difference between seats without numbers and seats with numbers? 
For seats without numbers, we have the formula $(n-1)!$. But how about seats with numbers? Does it make a difference?
 A: If the seats are not numbered, we only care about the relative order of the people.  Suppose Andrea is one of those people.  We seat her first, then count the number of distinguishable seating arrangements relative to Angela.  Once Angela has been seated, we can seat the other $n - 1$ people at the table in $(n - 1)!$ orders as we proceed clockwise around the table.  
If the seats are numbered, then a seating arrangement is distinguished by who sits in which numbered seat.  There are $n!$ orders in which we can line up the people, then seat them in seats $1$ through $n$.  In effect, numbering the seats is equivalent to arranging people in a row. 
A: The problem you're asking about sounds like it's the problem of distinctness or order. When counting the different ways a process may occur, or objects are arranged, whether or not we can tell the difference between our objects or the places they go matters a great deal.
In this example, we have two different questions:

How many ways can $n$ people be seated in $n$ distinct chairs?

to be contrasted with

How many ways can $n$ people be seated in $n$ chairs?

In the first question, given a seating of people, if we make two people swap places, we have a different arrangement, whereas in the second question we would have the same arrangement -- after all, as long as people are seated in chairs we don't differentiate between their order.
Therefore, we have the answers $(n-1)!$ and $1$ for each question, respectively.
A: If the seats are numbered, then it doesn't matter whether the seats are arranged in a straight row or in a circle or in some other set of random locations on the plane (such as in a rectangular grid, if $n$ is composite). The number of ways to seat $n$ distinct people in $n$ distinct seats is just $n!$, because there are $n$ ways to place the first person (in any one of the $n$ available seats), $n-1$ ways to place the second person (because the second person can be placed in any seat except where the first person was placed), and so on.
