In how many ways can 5 girls and 3 boys stand in a row so that no two boys are together ?

Options

A. A. 14400
B. B. 7200
C. C. 3600
D. D. 3200

Answer

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

### Step 1: Arrange the Girls
First, we need to arrange the 5 girls. The number of ways to arrange 5 girls is given by the factorial of the number of girls:

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

### Step 2: Identify the Positions for Boys
Once the girls are arranged, we can visualize the arrangement. The girls will create gaps where the boys can be placed. For 5 girls, there will be 6 possible gaps (one before each girl and one after the last girl):

- Gap 1 (before the first girl)
- Gap 2 (between the first and second girl)
- Gap 3 (between the second and third girl)
- Gap 4 (between the third and fourth girl)
- Gap 5 (between the fourth and fifth girl)
- Gap 6 (after the last girl)

### Step 3: Choose Gaps for Boys
We need to choose 3 gaps out of these 6 to place the boys, ensuring that no two boys are in the same gap. The number of ways to choose 3 gaps from 6 is given by the combination formula:

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

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

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

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

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

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

Thus, the total number of ways in which 5 girls and 3 boys can stand in a row such that no two boys are together is **14400**.