In how many ways can twelve girls be arranged in a row if two particular girls must occupy the end places

Options

A. `(10!)/(2!)`
B. `12!`
C. `10! * 2!`
D. `(12!)/(2!)`

Answer

To solve the problem of arranging twelve girls in a row with the condition that two particular girls must occupy the end places, we can follow these steps:

### Step 1: Fix the Positions of the Two Girls
Since two particular girls must occupy the end places, we can denote these girls as Girl A and Girl B. There are two possible arrangements for these two girls at the ends:
- Girl A at the left end and Girl B at the right end
- Girl B at the left end and Girl A at the right end

Thus, there are 2 ways to arrange the two particular girls at the ends.

**Hint:** Consider the different ways to arrange the two specific girls in the fixed positions.

### Step 2: Arrange the Remaining Girls
After placing the two particular girls in the end positions, we have 10 remaining girls to arrange in the middle positions. The number of ways to arrange these 10 girls is given by the factorial of the number of girls, which is 10!.

**Hint:** Remember that the number of arrangements of n distinct objects is given by n!.

### Step 3: Calculate the Total Arrangements
Now, we can find the total number of arrangements by multiplying the number of ways to arrange the two girls at the ends by the number of ways to arrange the remaining girls in the middle. Therefore, the total number of arrangements is:

Total arrangements = (Ways to arrange the two girls) × (Ways to arrange the remaining girls)
Total arrangements = 2 × 10!

### Step 4: Final Calculation
Now, we can compute the value of 10!:
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

Thus, the total arrangements become:
Total arrangements = 2 × 3,628,800 = 7,257,600

### Conclusion
The total number of ways to arrange the twelve girls in a row, with the condition that two particular girls occupy the end places, is **7,257,600**.

**Final Answer:** 7,257,600

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