Question 1114142
.
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.


<pre>
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.



<U>Answer</U>.  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.
</pre>

----------------
On permutations, and specifically on circular permutations, &nbsp;see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Introduction-to-Permutations.lesson>Introduction to Permutations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/PROOF-of-the-formula-on-the-number-of-permutations.lesson>PROOF of the formula on the number of Permutations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Problems-on-Permutations.lesson>Problems on Permutations</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Persons-sitting-around-a-circular-table.lesson>Persons sitting around a circular table</A> (*)


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/OVERVIEW-the-lessons-on-Permutations-and-Combinations.lesson>OVERVIEW of lessons on Permutations and Combinations</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic &nbsp;"<U>Combinatorics: Combinations and permutations</U>". 



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.