The number of ways of dividing 20 persons into 10 couples is

Options

A. `(20!)/(2^(10)10!)`
B. `""^(20)C_(10)`
C. `(20!)/((21!)^(9))`
D. (20!)/(2^(10)

Explanation

To find the number of ways to divide 20 persons into 10 couples, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to pair 20 individuals into 10 couples. Each couple consists of 2 individuals.

2. **Choosing Couples**:
- First, we choose 2 persons from the 20 to form the first couple. The number of ways to choose 2 persons from 20 is given by the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).
- For the first couple, the number of ways is:
\[ C(20, 2) = \frac{20!}{2!(20-2)!} = \frac{20!}{2! \cdot 18!} \]

3. **Continuing the Process**:
- After forming the first couple, we have 18 persons left. We then choose 2 from these 18 to form the second couple:
\[ C(18, 2) = \frac{18!}{2!(18-2)!} = \frac{18!}{2! \cdot 16!} \]
- We repeat this process until we have formed all 10 couples.

4. **Generalizing the Selection**:
- The total number of ways to form the couples can be expressed as:
\[ C(20, 2) \times C(18, 2) \times C(16, 2) \times \ldots \times C(2, 2) \]

5. **Calculating the Total**:
- This can be simplified as:
\[ \frac{20!}{(2!)^{10} \cdot 10!} \]
- Here, \( (2!)^{10} \) accounts for the fact that each couple can be arranged in 2 ways (e.g., (A, B) is the same as (B, A)), and \( 10! \) accounts for the arrangement of the 10 couples themselves.

6. **Final Result**:
- Therefore, the number of ways to divide 20 persons into 10 couples is:
\[ \frac{20!}{2^{10} \cdot 10!} \]

### Final Answer:
\[ \frac{20!}{2^{10} \cdot 10!} \]