Question 1134223
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(a) In how many ways can 6 people be arranged so that they form a circle between them; 
(b) and if 2 particular people of the 6 people quarrel, how many arrangements must be made if the quarrelsome pair 
must not stand next to each other?
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            Pay attention on how I edited your post to separate and clearly show that there are TWO QUESTIONS here.



<pre>
(a)  There are  {{{n!/n}}} = (n-1)!  circular arrangements of n objects.

     In this case n = 6, so there are (6-1)! = 5! = 120 circular arrangements of 6 persons.



(b)  Among these 120 circular arrangements, there are ONLY TWO, where the opponents are neighbors. 


    So, the answer to question (b) is  120 - 2 = 118 circular arrangements.
</pre>

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On circular arrangements, &nbsp;see the lesson

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