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.)
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.
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.