How many seating arrangements can George make if rotations and reflections of each seating arrangement are not considered different?

Question

I was wondering if I could get some help with this problem. I am a middle school student, so simpler explanations would be very much appreciated.

George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of each seating arrangement are not considered different? (Note: In this problem, if one seating is a reflection of another, then the two are considered the same!)

I know that there would be (8-1)! total arrangements for a round table where rotations are considered the same. Am I supposed to divide by 8 again to account for the reflections?

Answer 1

Seat George. His wife must be seated opposite him. That leaves six open seats. Line up the other women in some order. The first of them can be seated in six ways. Her husband sits opposite to her, leaving four choices for the next woman. Her husband sits opposite to her. That leaves two choices for the third woman. Her husband sits opposite to her. Therefore, up to rotation, there are $6 \cdot 4 \cdot 2$ possible seating arrangements. However, we have not accounted for reflections, so we must divide by $2$, which gives $$\frac{1}{2} \cdot 6 \cdot 4 \cdot 2 = 24$$ seating arrangements, in agreement with Henning Makholm's answer.

My thanks to Henning Makholm for pointing out the flaw in my initial attempt to solve the problem.

Answer 2

Let's number the chairs around the table:

     8 
 7       1

6         2

 5       3
     4

Since rotations don't matter, we can decide once and for all that George himself will sit in chair 8, and his wife therefore in chair 4.

Since reflections don't matter, we can decide once and for all that the person sitting in chair 2 must be a man.

Convince yourself that every allowed seating is a possibly-rotated-or-reflected version of exactly one seating that follows these two additional rule.

Now, we can start by choosing who sits in chair 1. There are 6 possibilities for that, and our choice will also implicitly put that person's spouse in chair 5.

Now there are 4 possibilities for who to sit in chair 3.

Finally only one man and his wife are left. By our decision the man must sit in chair 2, so there's no choice here.

All in all we have $$ 6 \times 4 = 24 $$ possible seatings.