SOLUTION: In how many ways can 4 gentlemen and 2 ladies be seated at a round table so that the ladies are not together? In how many of these ways will three particular gentlemen be next to e

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.