In how many ways can three couples be seated around a circular table on six equally spaced identical chairs such that for each couple, the husband and the wife occupy diametrically opposite chairs?

4

8

16

24

To solve this problem, we need to consider the seating arrangements around a circular table where each couple is seated with the husband and wife occupying diametrically opposite chairs.

### Steps to solve:

1. **Fixing one person to avoid rotational symmetry**:
   - Since the table is circular, any rotation of a seating arrangement results in an identical configuration. To eliminate equivalent rotations, we can fix one person in a specific position. Let's fix one person, say the husband of the first couple, in a particular seat.

2. **Arranging the spouse opposite the fixed person**:
   - Once the first person is fixed, their spouse must sit directly opposite. There's only one way to seat this person.

3. **Arranging the other couples**:
   - There are now 4 remaining seats, two of which are diametrically opposite. We can place the second husband in any of these two seats, which gives us 2 choices. 
   - After placing the second husband, his wife must sit opposite him, so there's only 1 way to place her.
   - The third couple has no choice but to sit in the remaining pair of opposite seats.

4. **Counting all the arrangements**:
   - We fixed one person at the start, but for any of the other couples, the husband and wife can swap their seats, leading to \(2 \times 2 \times 2 = 8\) possible configurations.

So, the total number of ways to seat the three couples is:

\[
\text{Total arrangements} = 2 \times 2 = 4 \times 2 = 8
\]

Therefore, the correct answer is **8**.

Would you like more details or have any questions about this? Here are some related questions you might find interesting:

1. How does the seating arrangement change if the chairs are not identical?
2. What if the couples cannot sit opposite each other? How many arrangements are possible?
3. How would the solution change if we had 4 couples instead of 3?
4. How many arrangements are possible if two specific people must sit next to each other?
5. How does the problem change if the couples must sit next to each other?

**Tip**: When solving circular permutation problems, always consider fixing one element to avoid overcounting due to rotational symmetry.

Explore the mathematical problem of seating three couples around a circular table such that each couple's husband and wife sit diametrically opposite. Understand the combinatorial approach and find the solution step-by-step. Suitable for high school students interested in combinatorics and circular permutations.

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