What counts as different
Around a round table there is no first seat and no end of the row. Two seatings are the same arrangement when everyone has the same neighbor on their left and the same neighbor on their right.
Turning everyone one seat round keeps all of those neighbors, so it gives the same arrangement. Swapping two people changes who sits next to whom, so it gives a new one.
The first two tables are one arrangement, turned a seat clockwise. In the third, B and C have swapped, so A now sits between C and D.
Dividing out the turns
In a line, 4 people have 4! = 24 orders. Read a table clockwise from each of its 4 seats in turn and you get 4 of those line orders, all the same arrangement at the table.
So the 24 line orders count every table arrangement 4 times, and there are 24 ÷ 4 = 6. In general, n people around a table make n! ÷ n = (n − 1)! arrangements.
Seating one person first
The same count comes another way. Seat A anywhere: every seat is like every other, so this choice changes nothing. But once A is seated, the turning is fixed, because each other seat is now "one to the left of A", "two to the left of A", and so on.
The other 3 people then fill those 3 seats in 3 × 2 × 1 = 6 ways. Written out with A at the top and reading clockwise, they are ABCD, ABDC, ACBD, ACDB, ADBC and ADCB.
Two who must sit together
Six people A to F sit at a round table, and B and C must sit next to each other. Glue B and C into one item. There are now 5 items around the table, which make (5 − 1)! = 4! = 24 arrangements.
Inside the glued item, B and C can sit as BC or CB, clockwise. Each of the 24 arrangements comes in both versions, so there are 24 × 2 = 48 with B and C together.
B and C are glued into one item in both tables. Swapping them inside the item gives a different arrangement, so the count of 24 is doubled.
Two who must sit apart
Every arrangement has B and C either together or apart, never both. So the arrangements with them apart are all of them minus those with them together: (6 − 1)! = 120 in all, and 120 − 48 = 72.
Counting the complement like this is usually quicker than counting "apart" directly, which would mean placing B and then counting the seats left for C that are not next to B.
Fixed seats, and necklaces
If A and B must sit directly opposite each other among six people, seat A first to fix the turning. B then has only one seat, and the other 4 people fill the remaining 4 seats in 4! = 24 ways.
Beads on a necklace or keys on a ring can also be turned over, which reverses the clockwise order. Then each arrangement and its mirror image are the same, and for 3 or more beads there are (n − 1)! ÷ 2. People at a table cannot be turned over, so there the count stays (n − 1)!.
The usual slips
Counting n! for a table: that is the count for a line, where turning everyone would give a different order.
Forgetting the swap inside a glued pair, which halves the answer, or ordering the glued items in a line, which multiplies it by the number of items.