The total number of ways in which four boys and four girls can be seated around a round table, so that no girls sit together is equal to

Options

A. 7!
B. 3!4!
C. 4!
D. 4!4!

Answer

To solve the problem of seating 4 boys and 4 girls around a round table such that no two girls sit together, we can follow these steps:

### Step-by-Step Solution:

1. **Fix One Boy**:
Since the arrangement is circular, we can fix one boy in one position to eliminate the effect of rotations. Let's fix Boy 1 (B1).

2. **Arrange the Remaining Boys**:
After fixing B1, we have 3 boys left (B2, B3, B4) to arrange. The number of ways to arrange these 3 boys is given by \(3!\):
\[ 3! = 6 \]

3. **Identify Spaces for Girls**:
Once the boys are seated, they create spaces for the girls. With 4 boys seated, there will be 4 gaps between them (one gap before each boy and one after the last boy). We can visualize it as follows:
- B1 _ B2 _ B3 _ B4 _
This gives us 4 available spaces for the girls.

4. **Arrange the Girls in the Gaps**:
We need to place the 4 girls (G1, G2, G3, G4) in these 4 gaps. Since we want to ensure that no two girls sit together, we can place one girl in each gap. The number of ways to arrange the 4 girls in the 4 gaps is given by \(4!\):
\[ 4! = 24 \]

5. **Calculate Total Arrangements**:
To find the total number of ways to arrange the boys and girls under the given conditions, we multiply the number of arrangements of boys by the number of arrangements of girls:
\[ \text{Total Arrangements} = 3! \times 4! = 6 \times 24 = 144 \]

### Final Answer:
The total number of ways in which 4 boys and 4 girls can be seated around a round table, so that no girls sit together, is **144**.