Number of ways of arranging 4 boys and 3 girls so that no boy is in between any two girls

Options

A. 360
B. 720
C. 180
D. 90

Answer

To solve the problem of arranging 4 boys and 3 girls such that no boy is in between any two girls, we can follow these steps:

### Step 1: Understand the arrangement
Since we need to ensure that no boy is placed between two girls, we can think of the girls as forming a block. This means we will treat the group of girls as a single unit.

### Step 2: Arrange the girls
We have 3 girls (let's call them G1, G2, and G3). The number of ways to arrange these 3 girls is given by the factorial of the number of girls:
\[ \text{Ways to arrange girls} = 3! = 6 \]

### Step 3: Create slots for boys
Once the girls are arranged, we can visualize the arrangement as follows:
- G1 G2 G3

This arrangement creates slots for the boys. The slots are:
- Before G1
- Between G1 and G2
- Between G2 and G3
- After G3

This gives us a total of 4 slots where we can place the boys.

### Step 4: Arrange the boys
We have 4 boys (let's call them B1, B2, B3, and B4). Since we can place the boys in any of the 4 slots, we can arrange the boys in these slots. The number of ways to arrange the 4 boys is given by:
\[ \text{Ways to arrange boys} = 4! = 24 \]

### Step 5: Combine the arrangements
The total number of arrangements is the product of the arrangements of the girls and the arrangements of the boys:
\[ \text{Total arrangements} = (\text{Ways to arrange girls}) \times (\text{Ways to arrange boys}) = 3! \times 4! = 6 \times 24 = 144 \]

### Final Answer
Thus, the total number of ways to arrange 4 boys and 3 girls such that no boy is in between any two girls is:
\[ \boxed{144} \]