In how many ways can 5 boys and 4 girls be seated at a round table, if no two girls are together?

Answer

To solve the problem of seating 5 boys and 4 girls at a round table such that no two girls are seated together, we can follow these steps:

### Step 1: Arrange the Boys
Since we are dealing with a circular arrangement, we can fix one boy's position to eliminate the effect of rotations. The remaining 4 boys can be arranged in the remaining seats.

The number of ways to arrange the 4 boys is given by:
\[ (5 - 1)! = 4! = 24 \]

### Step 2: Identify the Positions for Girls
Once the boys are seated, they create 5 gaps (or positions) where the girls can sit. These gaps are located between each pair of boys and at the ends.

### Step 3: Select Positions for the Girls
We need to choose 4 out of these 5 available gaps to seat the girls. The number of ways to choose 4 gaps from 5 is given by the combination:
\[ \binom{5}{4} = 5 \]

### Step 4: Arrange the Girls
After selecting the gaps, we can arrange the 4 girls in those chosen gaps. The number of ways to arrange the 4 girls is given by:
\[ 4! = 24 \]

### Step 5: Calculate the Total Arrangements
Now, we can find the total number of arrangements by multiplying the number of arrangements of boys, the number of ways to choose the gaps for the girls, and the arrangements of the girls:
\[ \text{Total arrangements} = (4!) \times \binom{5}{4} \times (4!) = 24 \times 5 \times 24 \]

Calculating this gives:
\[ 24 \times 5 = 120 \]
\[ 120 \times 24 = 2880 \]

### Final Answer
Thus, the total number of ways to seat 5 boys and 4 girls at a round table, ensuring that no two girls are seated together, is:
\[ \boxed{2880} \]