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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.
(a) There are = (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.
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On circular arrangements, see the lesson
- Persons sitting around a cicular table
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 lesson is 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.