# Circular permutations - confusion in textbook example

I have a confusion in one of the examples given in the book Discrete Mathematics: Elementary and Beyond. The exercise says:

In how many ways, in a party of six guests, can the people be seated around a circular table, given that Alice [the host] can too sit anywhere?

The answer given is a baffling $7.6.5 \; ... \;1 = 5040$

In the previous example, the host was asked to fix her place as it was her birthday (but the actual reason is to establish a point of reference, isn't it?). The answer then simply was $6!$, which makes sense. But now, this consideration of circular arrangements has been discarded and the answer matches that of arranging seven people in a straight line!

Is the answer wrong or am I missing something here?

• The convention, particularly when one uses the term circular permutation, is that if arrangement $\beta$ can be obtained from $\alpha$ by a rotation, the arrangements are to be viewed as the same. Of course they are not the same, one chair is nearest to the kitchen. But I would say that the conventional answer is $6!$. – André Nicolas Nov 16 '13 at 4:27
• @AndréNicolas Yes, that is a very interesting point! I always felt intuitively that arrangement in circle and arrangement in line felt similar ... too similar, in fact. But such subtleties (or should I say trivialities) are not part of mathematical theory. But where does that leave beginners like me? Is the book, highly acclaimed and published by Springer, misleading unfortunate students like me? :P – ankush981 Nov 16 '13 at 4:32
• There is ambiguity in the statement of the problem, it is not said that arrangements that differ by a rotation are to be viewed as the same, and it doesn't say they aren't. Let us hope that any exam question will be unambiguous. One can always protect oneself by giving the answer under each interpretation. – André Nicolas Nov 16 '13 at 4:39
• @AndréNicolas Thanks for answering. No, exams won't be a problem as I'm doing this as a recreational activity. :) – ankush981 Nov 16 '13 at 5:02
• @AndréNicolas (or dotslash), would you mind posting the comment/explanation as an answer so that this question doesn't remain unanswered? – Mark S. Dec 15 '13 at 15:35

Answered by André Nicolas in comments above: "The convention, particularly when one uses the term circular permutation, is that if arrangement $\beta$ can be obtained from $\alpha$ by a rotation, the arrangements are to be viewed as the same. Of course they are not the same, one chair is nearest to the kitchen. But I would say that the conventional answer is $6!$."