Seating people in a circular table It has always been an interesting question. 

If we have $10$ chairs and a round table, how many ways are there of seating $10$ people? 

I would say there are $10!$ ways to seat the people due to there being $10$ choices for the first $9$ for the second?
But people often say the number of ways is: $(n-1)!$, but which one is correct and why?
 A: There are $6! = 720$ ways of seating six people in six chairs, BUT, when people speak of a "circular table" in this context, what they often have in mind is that the following two arrangements are the same since everyone has the same person to its left and the same person to its right in both arrangements.
$$
\begin{array}{cccccccccc}
& & A \\
& \nearrow & & \searrow \\
F & & & & B \\[6pt]
\uparrow & & & & \downarrow \\
E & & & & C \\
& \nwarrow & & \swarrow \\
& & D
\end{array}
$$
$$
\vphantom{frac\int\int}
$$
$$
\begin{array}{cccccccccc}
& & F \\
& \nearrow & & \searrow \\
E & & & & A \\[6pt]
\uparrow & & & & \downarrow \\
D & & & & B \\
& \nwarrow & & \swarrow \\
& & C
\end{array}
$$
This is a cyclic shift.  One speaks of "modding out by cyclic shifts", which means two things are considered identical if the only difference between them is a cyclic shift.
A: Since it is a round table, and the seats are identical, it doesn't matter if everyone moves one seat to the right or the left. So we fix one person on one chair, and we have 9 chairs left to put 9 people on for $9!$ ways. 
