If there are n people then they can be seated in a straight line in n! ways but if they are to be seated in a circle then there are (n-1)! ways. Why is this so? Can't we just treat a circle as a line? I mean that the boundary, if straightened, of a circle is just a line? Could someone please clear this confusion? Thanks.
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The cyclic permutations in the line seems equals in the circle – DiegoMath Oct 04 '20 at 19:56
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@DiegoMath I am sorry I don't understand your comment. Can you please explain it? – ryan1 Oct 04 '20 at 20:04
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The permutations $1,2,3,4$ and $4,1,2,3$, when bent into a circle, is the same (under possibly some rotation) – player3236 Oct 04 '20 at 20:07
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1By convention, in a circular permutation, we only care about the relative order of the objects. Suppose one of the (n) people is Mary. We can use her as a reference point. In how many ways can the remaining (n - 1) people be arranged in a circle as we proceed clockwise from Mary? – N. F. Taussig Oct 04 '20 at 20:10
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1Does this answer your question? Explanation circular permutation – N. F. Taussig Oct 04 '20 at 20:11
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https://math.stackexchange.com/questions/2387149/explanation-circular-permutation/2395407 – Lion Heart Oct 04 '20 at 20:34
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For example, if you want to sit 5 persons(for simplicity every person is a letter; A, B, C, D, E) in a line of chairs, you can order them in 5! ways i.e 120 different ways. So if we are thinking of a circular permutation problem we can solve it similar to a linear permutation. First, you have to choose someone to be the fixed position, and then arrange the remaining persons in a different order. Suppose that ms. A goes in the first chair. You have left 4 choices for the second chair, 3 for the third, 2 for the fourth, and 1 for the fifth. That means that if you chose one person for the first chair you have 4! ways of arranging the persons in the remaining four positions of the chairs. So the number of ways of arranging A, B, C, D, E in a circle of chairs is 1x4!=24. Note that (n-1)!=n!/n. In this case we have (5-1)!=4!=24=120/5
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