In how many ways 10 boys and 5 girls can sit around a circular table so that no two girls sit together.

Answer

To solve the problem of how many ways 10 boys and 5 girls can sit around a circular table such that no two girls sit together, we can follow these steps:

### Step 1: Arrange the Boys
Since we are dealing with a circular arrangement, we can fix one boy in position to eliminate the effect of rotations. This means we will arrange the remaining 9 boys around the table.

The number of ways to arrange \( n \) people in a circle is given by \( (n-1)! \). Therefore, the number of ways to arrange 10 boys is:
\[ (10-1)! = 9! \]

### Step 2: Identify Spaces for Girls
Once the boys are seated, they create spaces for the girls. In a circular arrangement of 10 boys, there are 10 gaps (or spaces) between them where the girls can sit.

### Step 3: Choose Spaces for Girls
We need to choose 5 out of these 10 spaces to place the girls, ensuring that no two girls sit together. The number of ways to choose 5 spaces from 10 is given by the combination formula \( \binom{n}{r} \):
\[ \binom{10}{5} \]

### Step 4: Arrange the Girls
After selecting the spaces, we can arrange the 5 girls in the chosen spaces. The number of ways to arrange 5 girls is given by \( 5! \).

### Step 5: Combine the Results
Now, we can combine all the results to find the total number of arrangements. The total number of ways to arrange the boys and girls is:
\[ 9! \times \binom{10}{5} \times 5! \]

### Final Calculation
Now we can calculate the values:
- \( 9! = 362880 \)
- \( \binom{10}{5} = \frac{10!}{5! \times 5!} = 252 \)
- \( 5! = 120 \)

Thus, the total number of ways is:
\[ 9! \times \binom{10}{5} \times 5! = 362880 \times 252 \times 120 \]

Calculating this gives:
\[ 362880 \times 252 = 91545600 \]
Then,
\[ 91545600 \times 120 = 10985472000 \]

So, the final answer is:
\[ \text{Total ways} = 10985472000 \]