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 1: Arrange the Boys
Since the arrangement is around a round table, we can fix one boy's position to eliminate the effect of rotations. The remaining 3 boys can then be arranged in the remaining seats.

- The number of ways to arrange 4 boys in a round table is given by:
\[ (n - 1)! = (4 - 1)! = 3! = 6 \]

### Step 2: Identify Spaces for Girls
After seating the boys, there will be spaces between them where the girls can sit. For 4 boys seated around the table, there will be 4 gaps created (one between each pair of boys).

- The arrangement looks like this: B1 _ B2 _ B3 _ B4 _ (where "_" represents a gap for a girl).

### Step 3: Arrange the Girls
Now, we need to arrange the 4 girls in these 4 gaps. Since we want to ensure that no two girls sit together, each girl must occupy a separate gap.

- The number of ways to arrange 4 girls in the 4 gaps is given by:
\[ 4! = 24 \]

### Step 4: 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:

\[ \text{Total arrangements} = \text{Arrangements of boys} \times \text{Arrangements of girls} = 3! \times 4! = 6 \times 24 = 144 \]

### Final Answer
Thus, 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**.

---