To figure out most combinatorial problems, you need to establish a “construction plan” that tells you how to build all the combinations with the specified properties. While making that plan, you have to ensure three things:
will my plan build all the valid combinations ? If no, modify the plan.
will my plan build only valid combinations ? if no, modify the plan.
will my plan build some combinations more than once ? if yes, either modify the plan or figure out the number of duplicates.
Once this is done, you usually just count the different choices that are left free in the construction plan…
Let’s apply this to the problem, naming A and B the two women who cannot be placed next to each other. I propose the following plan:
first let’s place the 9 women in a row:
place A somewhere: 9 choices.
place B: 6 choices, since once A is placed, it cancels two chairs (if A is in first chair, then it cancels chair 2 and 9…)
place all the other women randomly: 7! choices.
then place the aligned women around the table, starting at a fixed particular chair and turning clockwise: only one manner to do that.
Now:
does that plan build all (valid) combinations: yes, because whatever valid placement around the table can just be “unrolled” in a row, and obviously we have built every possible row.
does the plan build only valid combinations: yes, we took care of A and B
does that plan build some arrangements several times : yes, when placing the row around the table, we could have started at any of the 9 chairs and this would have yield the same (relative) positions around the table. So the plan produces the same arrangement 9 times.
Conclusion:
There are 9*6*7!/9 arrangements.
Remarks;
one could consider two arrangements that just differ from the way they turn (clockwise or anti-clockwise) to be equivalent (since every person has the same neighbors)… in that case divide the result by 2…
the construction plan proposed here is not the only possible, nor is it the “best”… I just came up with it while answering. Everyone has its own inner logic when it comes to building arrangements… sometimes it is good to try a completely different plan to see if it ends up with the same result. This helps to educate intuition.