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?

To solve the problem of determining the number of ways three couples can be seated around a circular table such that each couple occupies diametrically opposite chairs, we can break down the problem as follows:

### Step 1: Fix one person to avoid rotational symmetry
Since the chairs are arranged in a circle, seating arrangements that can be obtained by rotation are considered identical. To eliminate this symmetry, we can fix one person in a specific seat. Let's fix one of the husbands (say Husband A) in a particular seat.

### Step 2: Seat the spouse of the fixed person
Given that each couple must sit diametrically opposite to each other, Wife A must occupy the seat directly opposite Husband A. This arrangement is fixed.

### Step 3: Arrange the remaining couples
After seating Husband A and Wife A, there are two seats left on one side of the table and two seats on the opposite side. The remaining two husbands can be seated in the two remaining seats on one side in \(2!\) (2 factorial) ways. After seating the two remaining husbands, their wives must occupy the diametrically opposite seats.

### Step 4: Count the total number of arrangements
Since Husband A's seat was fixed initially to remove rotational symmetry, we have already accounted for all possible unique arrangements. The total number of arrangements is thus:

\[
2! \times 2! = 2 \times 2 = 4
\]

### Final Answer
There are **4** distinct ways to seat the three couples around the table under the given conditions.

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Would you like more details on the solution, or do you have any other questions?

Here are 5 related questions to explore further:

1. What if the table had 8 seats and 4 couples, maintaining the same condition?
2. How does the solution change if the seating order around the table matters?
3. What is the number of ways to arrange 3 couples if only one couple must sit opposite each other?
4. How many ways can the couples be arranged linearly if no condition is imposed?
5. How would the problem change if each couple must sit adjacent to each other instead?

**Tip:** For problems involving circular permutations, fixing one element is a common technique to account for rotational symmetry.

Explore the mathematical solution to counting the number of ways three couples can be seated around a circular table such that each couple occupies diametrically opposite chairs. Understand the concept of circular permutations and factorial calculations in combinatorics.

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