The number of ways of dividing `15` men and `15` women into `15` couples each consisting a man and a woman is

Options

A. 1960
B. `15!`
C. `(15!)^(2)`
D. `14!`

Answer

To solve the problem of dividing 15 men and 15 women into 15 couples, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to pair 15 men with 15 women such that each couple consists of one man and one woman.

2. **Choosing Couples**:
- For the first couple, we have 15 choices for the man and 15 choices for the woman. Thus, the number of ways to choose the first couple is \(15 \times 15 = 15^2\).
- After choosing the first couple, we have 14 men and 14 women left. For the second couple, the number of ways to choose is \(14 \times 14 = 14^2\).
- This pattern continues until we reach the last couple, where we have 1 man and 1 woman left, giving us \(1 \times 1 = 1^2\).

3. **Total Number of Ways**:
- Therefore, the total number of ways to form all couples can be expressed as:
\[ 15^2 + 14^2 + 13^2 + \ldots + 1^2 \]
- This is the sum of squares of the first 15 natural numbers.

4. **Using the Formula for Sum of Squares**:
- The formula for the sum of squares of the first \(n\) natural numbers is:
\[ \text{Sum} = \frac{n(n+1)(2n+1)}{6} \]
- For \(n = 15\):
\[ \text{Sum} = \frac{15 \times 16 \times 31}{6} = 1240 \]

5. **Final Calculation**:
- However, we need to consider that each arrangement of couples can be done in \(15!\) ways (since the order of couples matters).
- Therefore, the total number of ways to form the couples is:
\[ 15! \times 15! \]

6. **Conclusion**:
- The number of ways to divide 15 men and 15 women into 15 couples is:
\[ (15!)^2 \]

### Final Answer:
The number of ways of dividing 15 men and 15 women into 15 couples is \( (15!)^2 \).

---