What is the number of ways that 4 boys and 3 girls can be seated so that boys and girls alternate ?

Options

A. 12
B. 72
C. 120
D. 144

Answer

To solve the problem of seating 4 boys and 3 girls alternately, we can follow these steps:

### Step 1: Determine the seating arrangement
Since we have more boys than girls, the only possible arrangement that allows for alternating seating is:
- Boy, Girl, Boy, Girl, Boy, Girl, Boy

This means the arrangement starts and ends with a boy.

### Step 2: Arrange the boys
We have 4 boys, and they can be seated in the 4 boy positions. The number of ways to arrange 4 boys is given by the factorial of the number of boys:
- Number of arrangements for boys = 4! = 4 × 3 × 2 × 1 = 24 ways

### Step 3: Arrange the girls
We have 3 girls, and they can be seated in the 3 girl positions. The number of ways to arrange 3 girls is given by the factorial of the number of girls:
- Number of arrangements for girls = 3! = 3 × 2 × 1 = 6 ways

### Step 4: Calculate the total arrangements
To find the total number of arrangements where boys and girls alternate, we multiply the number of arrangements for boys by the number of arrangements for girls:
- Total arrangements = (Number of arrangements for boys) × (Number of arrangements for girls)
- Total arrangements = 24 × 6 = 144 ways

### Final Answer
Thus, the total number of ways that 4 boys and 3 girls can be seated so that boys and girls alternate is **144 ways**.

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