In how many different ways can 4 boys and 3 girls be arranged in a row such that all the boys stand together and all the girls stand together ?

Options

A. 75
B. 576
C. 288
D. 24

Explanation

To solve the problem of arranging 4 boys and 3 girls in a row such that all boys stand together and all girls stand together, we can follow these steps:

### Step-by-Step Solution:

1. **Treat Boys and Girls as Groups**: Since all boys must stand together and all girls must stand together, we can treat the group of boys as a single unit (block) and the group of girls as another single unit (block). Therefore, we have two blocks to arrange: one block of boys and one block of girls.

2. **Arrange the Blocks**: The two blocks (Boys block and Girls block) can be arranged in a row in 2! (factorial of 2) ways. This is because we can either have the Boys block first followed by the Girls block, or the Girls block first followed by the Boys block.

\[ \text{Ways to arrange blocks} = 2! = 2 \]

3. **Arrange the Boys within their Block**: Within the Boys block, the 4 boys can be arranged among themselves in 4! ways.

\[ \text{Ways to arrange boys} = 4! = 4 \times 3 \times 2 \times 1 = 24 \]

4. **Arrange the Girls within their Block**: Similarly, within the Girls block, the 3 girls can be arranged among themselves in 3! ways.

\[ \text{Ways to arrange girls} = 3! = 3 \times 2 \times 1 = 6 \]

5. **Calculate the Total Arrangements**: To find the total number of arrangements, we multiply the number of ways to arrange the blocks by the number of arrangements of boys and girls within their respective blocks.

\[ \text{Total arrangements} = (\text{Ways to arrange blocks}) \times (\text{Ways to arrange boys}) \times (\text{Ways to arrange girls}) \]

\[ \text{Total arrangements} = 2! \times 4! \times 3! = 2 \times 24 \times 6 \]

6. **Perform the Calculation**:

\[ 2 \times 24 = 48 \]
\[ 48 \times 6 = 288 \]

Thus, the total number of different ways to arrange 4 boys and 3 girls in a row such that all boys stand together and all girls stand together is **288**.