The number of ways in which 5 boys and 3 girls can be seated in a row, so that no two girls sit together is

Options

A. `|ul(8)`
B. `|ul(5)xx|ul(3)`
C. `|ul(3)xx.^(5)P_(4)`
D. `|ul(5)xx.^(6)P_(3)`

Answer

To solve the problem of seating 5 boys and 3 girls in a row such that no two girls sit together, we can follow these steps:

### Step 1: Arrange the Boys
First, we need to arrange the 5 boys. The number of ways to arrange 5 boys in a row is given by the factorial of the number of boys:

\[ \text{Ways to arrange boys} = 5! = 120 \]

### Step 2: Identify the Positions for Girls
Once the boys are arranged, we can identify the possible positions where the girls can sit. When 5 boys are seated, they create 6 potential gaps for the girls to occupy (one before the first boy, one between each pair of boys, and one after the last boy).

The arrangement looks like this:
- _ B _ B _ B _ B _ B _

Here, each underscore (_) represents a potential position for a girl. Thus, we have 6 gaps.

### Step 3: Choose Positions for the Girls
We need to choose 3 out of these 6 gaps to place the girls. The number of ways to choose 3 gaps from 6 is given by the combination formula:

\[ \text{Ways to choose gaps} = \binom{6}{3} \]

Calculating this:

\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 4: Arrange the Girls
After choosing the gaps, we can arrange the 3 girls in those selected gaps. The number of ways to arrange 3 girls is given by:

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

### Step 5: Calculate Total Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to arrange the boys, the number of ways to choose the gaps, and the number of ways to arrange the girls:

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

Substituting the values we calculated:

\[ \text{Total arrangements} = 120 \times 20 \times 6 = 14400 \]

### Conclusion
The total number of ways in which 5 boys and 3 girls can be seated in a row such that no two girls sit together is:

\[ \boxed{14400} \]