The total number of ways in which 4 boys and 4 girls can form a line, with boys and girls alternating, is

Options

A. (4!)2
B. 8!
C. 2(4!)^2
D. `4!.""^(5)P_(4)`

Answer

To solve the problem of arranging 4 boys and 4 girls in a line such that they alternate, we can break it down into a few steps.

### Step-by-Step Solution:

1. **Identify the Arrangements**:
There are two possible arrangements for alternating boys and girls:
- Case 1: Boy-Girl-Boy-Girl-Boy-Girl-Boy-Girl
- Case 2: Girl-Boy-Girl-Boy-Girl-Boy-Girl-Boy

2. **Calculate Arrangements for Case 1**:
- In Case 1, we have 4 boys and 4 girls.
- The boys can be arranged in their positions in \(4!\) (factorial of 4) ways.
- The girls can also be arranged in their positions in \(4!\) ways.
- Therefore, the total arrangements for Case 1 = \(4! \times 4!\).

3. **Calculate Arrangements for Case 2**:
- Similarly, in Case 2, the arrangement is the same as Case 1.
- The boys can again be arranged in \(4!\) ways and the girls in \(4!\) ways.
- Therefore, the total arrangements for Case 2 = \(4! \times 4!\).

4. **Combine the Cases**:
- Since both cases are mutually exclusive, we add the total arrangements from both cases.
- Total arrangements = Arrangements from Case 1 + Arrangements from Case 2 = \(4! \times 4! + 4! \times 4!\).
- This simplifies to \(2 \times (4! \times 4!)\).

5. **Calculate \(4!\)**:
- \(4! = 4 \times 3 \times 2 \times 1 = 24\).

6. **Final Calculation**:
- Now, substituting \(4!\) into the total arrangements:
- Total arrangements = \(2 \times (24 \times 24) = 2 \times 576 = 1152\).

### Conclusion:
The total number of ways in which 4 boys and 4 girls can form a line, with boys and girls alternating, is **1152**.