A group of 2n students consisting of n boys and n girls are to be arranged in a row such that adjacent members are of opposite sex. The number of ways in which this can be done is

Options

A. `2(n!)`
B. `(n!)^(2)`
C. `2(n!)^(2)`
D. `n !`

Answer

To solve the problem of arranging 2n students (n boys and n girls) in a row such that adjacent members are of opposite sex, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement**:
- We need to arrange n boys and n girls in a way that no two boys or two girls are adjacent. This means the arrangement must alternate between boys and girls.

2. **Choosing Positions**:
- We can choose to place boys in the odd positions and girls in the even positions or vice versa.
- For example, if we have positions labeled as 1, 2, 3, ..., 2n, we can place boys in positions 1, 3, 5, ..., (2n-1) and girls in positions 2, 4, 6, ..., (2n).

3. **Counting the Positions**:
- There are n odd positions available for boys and n even positions available for girls.

4. **Arranging the Boys**:
- The n boys can be arranged among themselves in n! (n factorial) ways.

5. **Arranging the Girls**:
- Similarly, the n girls can also be arranged among themselves in n! ways.

6. **Calculating Total Arrangements**:
- Since the arrangements of boys and girls are independent, we multiply the number of arrangements of boys by the number of arrangements of girls.
- Therefore, the total number of arrangements is given by:
\[ \text{Total arrangements} = n! \times n! = (n!)^2 \]

7. **Final Answer**:
- The total number of ways to arrange the 2n students such that no two boys or two girls are adjacent is \((n!)^2\).