Mmmm. With all due respect, I'm not sure why you find those two statements contradictory. But let me address them separately.
Statement 1: "... your solution above (see Post #5) is really just the number of permutations of 3 things. The "circular" arrangement (once the 1 is placed) is irrelevant. You may as well just list the 6 permutations as 432, 342, 423, 243, 324, 234."
I think you agree with this, because your solution to the original post (where there are 8 people with one already seated) is 7! applying the same logic.
Your post #7 says:
Actually this is not a circular arrangement. If we seat eight people at a circular table can be done in (8-1)!=7! ways. That is true of any circular permutation. However, if there is an assigned seat, as in this case, that makes it an ordered table as opposed to an unordered table. That means there are seven other people left to be seated which can be done in 7! ways.
So here you are saying that the number of ways of sitting 8 people at a table (with no restrictions) is the same as the number of ways of sitting 8 people at a table where one person has an assigned seat (ie sitting 7 people at a table when one person is already in an assigned seat).
That seems contradictory to me.
I do agree with the answer (ie 7!), which is just the number of permutations of 7 things.
Statement 2: "BBB, I agree . I've always questioned the wording and answers to those typical textbook questions on circular arrangements around a table for the exact reason you mention (see post #8)."
I suppose this comes down to the definition of "permutation" and "arrangement" which are often used interchangeably (incorrectly in my opinion) in textbook questions.
Your post #10 says:
Now any talk of a difference in a round table over-against a ring is that of an utter armature.
When a table is mentioned, it is in a context, eg in a room. It is not merely theoretical any longer. Consider the following scenario of a table in a room with a window.
I could concede that the "permutation" (order) of seats around the table is the same - in both cases for example, B is to the left of A, etc.
BUT, I would argue that the "arrangement" of seats around the table is
not the same. In the first scenario, A has his/her back to the window while in the second scenario A has a view out the window. If you were person A and didn't want the bright light coming in at your face (in scenario 2), then you would prefer scenario 1. Because one is preferable to the other, they must be different.
If, however, they were literally rings with an Amethyst, a Birthstone, a Cubic zirconia and a Diamond, then they
would be the same, simply because of the rotatability of the ring.
I don't think what I've said is of "an utter armature" (or even amateur). I also don't think it is "such gross miss-information" but is a good topic for discussion.