The problem asks for the number of ways to arrange 6 pairs of married couples, where each couple sits together, in a circular arrangement of 12 chairs.

Step-by-step solution:

1. Treat each couple as a single unit: Since the problem specifies that each couple must sit next to each other, we can think of each couple as a "block" or "unit." This reduces the problem to arranging 6 units (the 6 couples) in a circular fashion.

2. Circular permutation of 6 units: In a circular arrangement, the number of ways to arrange $n$ objects is $(n-1)!$. Therefore, the number of ways to arrange the 6 couples (units) in a circle is: $(6-1)! = 5!$

3. Arranging individuals within each couple: Within each "couple block," the husband and wife can switch seats, which means there are 2 ways to arrange each couple. Since there are 6 couples, the total number of ways to arrange the individuals within the blocks is: $2! \times 2! \times 2! \times 2! \times 2! \times 2! = (2!)^6$

4. Final formula: The total number of ways to arrange the 6 couples in a circular arrangement, with each couple sitting together, is the product of the circular arrangement of the blocks and the internal arrangement of individuals within each block: $5! \times (2!)^6$

Answer:

The correct option is $\boxed{D}$.

Would you like further clarification or more details? Here are some related questions:

1. How would the answer change if the chairs were arranged in a straight line instead of a circle?
2. What is the significance of dividing by $n$ in circular permutations?
3. How would the problem change if some couples do not need to sit together?
4. Can this arrangement problem be solved using a different method, such as combinatorics without treating each couple as a unit?
5. What if the total number of couples were different, say 5 couples instead of 6?

Tip: In circular permutations, the formula $(n-1)!$ accounts for rotational symmetry, meaning that rotations of the same arrangement are considered equivalent.