I am having problem in understanding this concept:
Circular permutation : The definition in my book goes like that ' Arrangements of things in a circle or a ring are called circular permutations. The fundamental difference between linear and that of circular permutation is that in the former, there are always two separate ends but in circular permutations we cannot distinguish the two ends. For this,in linear permutations ,arrangements depend on the absolute position while in the case of circular permutations,we shall be concerned with relative positions of the things. Thus, no. of circular permutations of $n$ different things taken all at a time is $$(n-1)!$$ ways taking one of the $n$ things fixed.
This is what is written in the book. Now, it gave two questions ; 1. In how many ways can 6 boys form a ring? . The answer was,according to the formula, was $5!$ ways. 2. In how many ways can 6 men be seated at a round table? The answer, I thought,would be again $5!$ . But book gave the answer $6!$ giving reason that since $6$ men were to be arranged with respect to the table and not with themselves,hence the problem is equivalent to linear permutation.
But I didn't understand their reasoning; 6 men were forming a ring and sitting around a round table,what is the difference? And what the book wanted to tell by saying relative position ?
Another problem is the necklace problem: In how many ways can 6 beads of different colours be arranged to form necklace? The answer,again to me, was $5!$ . But the correct answer ,the book said,was $\frac{5!}{2}$ reasoning that clockwise & anti-clockwise arrangement can't be distinguished. But I couldn't understand this brief reasoning.
Plz help explaining these problems.
Best Answer
The "ring" question asks "how many ways are there to put 6 people in a ring", where two "ways" are identical if each individual has the same left and right neighbours.
The "table" question asks "how many ways are there to put 6 people at a table", where two "ways" are identical if each individual has the same left and right neighbours and the same place at the table.
It is easy to see that since the possible rotations are 6, the answer to the first question will be 6 times the answer to the second question, and indeed 6! = 6 x 5!.
Concerning the "necklace" problem, the difference from the "ring" problem is that you cannot put people upside down without noticing - while you can to so with a necklace, obtaining a pattern which is symmetric. So the number of combination is halved, because two symmetric combinations are considered as one.