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In how many ways can $2$ teachers and $4$ students, a total of 6 people, be seated around a circular table based on the following conditions:

(1) Teachers are seated opposite to each other (2) Teachers are seated next to each other

Here are my answers (1) Since the teachers must be seated opposite to each other, let the two opposite chairs be fixed for the teachers. There are $2! = 2$ ways to arrange the two teachers opposite to each other. There are $4! = 24$ ways for the students to arrange themselves.

Hence, there will be $(2 × 24)=48$ ways so that the teachers are seated opposite to each other.

(2) There are $2! = 2$ ways to arrange the two teachers seated next to each other. There are $4! = 24$ ways for the students to arrange themselves.

Hence, there will be $(2 × 24) = 48$ ways so that the teachers are seated next to each other.

I have the same result for the two questions, I am not sure if my answers are correct. Any comments or suggestions will be much appreciated. Thank you in advance.

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  • $\begingroup$ Do we count rotation of the table to be the same? If they are different, there are $6$ ways to arrange teacher for (1), $12$ ways for (2). $\endgroup$ Oct 20 '20 at 1:02
  • $\begingroup$ For two teachers sitting opposite each other, there are either six ways or one way how they can sit, depending if you count rotation or not. For two teachers sitting side by side, there are 12 or 2 ways. $\endgroup$
    – gnasher729
    Oct 20 '20 at 1:04
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The answer for the first part is actually only $24$: there's only one way to arrange the two teachers up to rotation of the table, so you counted each position twice. The second part's answer is indeed $48$.

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