.
To solve this problem, the best way (and I think, the standard way) is to consider the whole set of permutations and
the complementary set of permutations, and then to take the difference.
1. The whole set of permutations in this problem is the set of all permutations (seating arrangements) of 6 persons at
a round table without considering gender.
It is well known fact that the number of all such permutations (circular permutations) is (6-1)! = 5! = 120.
2. The complementary set of permutations is the set, where two ladies are sitting / (seating ?) together.
By considering this pair as one object, we have then the set of all circular permutations of 5 objects,
which consists of (5-1)! = 4! = 24 permutations.
We then must double this number 2*24 = 48 to distinct permutations of the type (Alice-Beatrice) and (Beatrice-Alice) inside these pairs.
It gives the final answer 120 - 48 = 72.
Answer. In how many ways can 4 gentlemen and 2 ladies be seated at a round table so that the ladies are not together? - in 72 wys.
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On permutations, and specifically on circular permutations, see the lessons
- Introduction to Permutations
- PROOF of the formula on the number of Permutations
- Problems on Permutations
- Persons sitting around a circular table (*)
- OVERVIEW of lessons on Permutations and Combinations
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic "Combinatorics: Combinations and permutations".
Save the link to this textbook together with its description
Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson
into your archive and use when it is needed.