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.)

Guest Jan 3, 2020

#2**0 **

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.

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

.Guest Jan 3, 2020

#3**+2 **

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.

USAMO2019 Jan 4, 2020