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# Confused

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In how many ways can 4 teachers and 4 students be seated at a circular table if each student sits directly between two teachers? (Two seatings are considered the same if one can be rotated to form the other.)

Jan 3, 2020

#1
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There are 4!*4! = 576 ways of arranging the teachers and students.

Jan 3, 2020
#2
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Fix one teacher T_1 at the top of the circle. T represents teachers, and 's' students. Find the ways to choose the students and teachers.

\(T_1,S_1,T_2,S_2,T_3,S_3,T_4,S_4\)

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Jan 3, 2020
#3
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First, we can fix the spot for where the teachers go. Then, we can simply plug in places for where the students can go. Note that there is 4 spaces for the students to sit. Thus, there is 4! ways to seat the students. In addition, there are 4! ways to order the teachers. Thus, there is 24*24 ways to order them. HOWEVER, the question states that you are ordering them in a circular table, meaning that you must divide by 8 to account for the number of rotations and reflections. Thus, our answer is 24*24/8, giving us 72.

Jan 4, 2020