https://file.aiquickdraw.com/mathbot/file/1728530801905fen9d6v3.jpg

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.

Terdapat 6 pasang suami istri akan makan bersama dengan duduk pada 12 kursi yang disusun melingkar. Banyak cara duduk dengan setiap pasangan suami istri duduk bersebelahan adalah ...

This problem involves arranging 6 couples in 12 seats arranged in a circle, with each couple sitting together. Learn the solution using permutation principles, including the circular permutation theorem and the arrangement of pairs.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation