Answer these questions independently of each other.
In how many ways can 12 soldiers stand in three queue such that one queue has 3 soldiers, another queue has 4 soldiers and so the remaining rows has 5 soldiers?

Options

A. `^12P_5xx^7P_4xx^3P_3`
B. `^12P_3xx^9P_4xx^5P_5`
C. `^12P_3xx^9P_4xx^5P_5xx3!`
D. `^12P_3xx^12P_4xx13P_5xx3!`

Answer

To solve the problem of arranging 12 soldiers into three queues with specific numbers of soldiers in each queue (3 in the first queue, 4 in the second queue, and 5 in the third queue), we can follow these steps:

### Step-by-Step Solution:

1. **Choose Soldiers for the First Queue**:
We need to select 3 soldiers from the total of 12 soldiers for the first queue. The number of ways to choose 3 soldiers from 12 is given by the permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \).
\[ \text{Ways to choose 3 soldiers} = P(12, 3) = \frac{12!}{(12-3)!} = \frac{12!}{9!} = 12 \times 11 \times 10 = 1320 \]

**Hint**: Use the permutation formula to select a specific number of items from a larger set.

2. **Choose Soldiers for the Second Queue**:
After selecting 3 soldiers, we have 9 soldiers left. Now, we need to choose 4 soldiers for the second queue.
\[ \text{Ways to choose 4 soldiers} = P(9, 4) = \frac{9!}{(9-4)!} = \frac{9!}{5!} = 9 \times 8 \times 7 \times 6 = 3024 \]

**Hint**: Remember to adjust the total number of soldiers available after each selection.

3. **Choose Soldiers for the Third Queue**:
After selecting soldiers for the first two queues, we have 5 soldiers left, all of whom will go into the third queue.
\[ \text{Ways to choose 5 soldiers} = P(5, 5) = \frac{5!}{(5-5)!} = 5! = 120 \]

**Hint**: When selecting all remaining items, the number of permutations is simply the factorial of the total number.

4. **Account for the Arrangement of Queues**:
The soldiers in each queue can be arranged among themselves. Therefore, we need to multiply the total arrangements by the factorial of the number of queues (3!).
\[ \text{Arrangement of queues} = 3! = 6 \]

**Hint**: Factorial represents the number of ways to arrange items, and it should be applied when the order matters.

5. **Calculate the Total Arrangements**:
Now, we combine all the calculations:
\[ \text{Total arrangements} = P(12, 3) \times P(9, 4) \times P(5, 5) \times 3! \]
\[ = 1320 \times 3024 \times 120 \times 6 \]

**Hint**: Multiply all the calculated values to find the final answer.

6. **Final Calculation**:
Performing the final multiplication:
\[ 1320 \times 3024 = 3991680 \]
\[ 3991680 \times 120 = 478401600 \]
\[ 478401600 \times 6 = 2870410560 \]

Therefore, the total number of ways to arrange the soldiers in the specified queues is **2870410560**.

### Final Answer:
The total number of ways 12 soldiers can stand in three queues (3 soldiers in the first, 4 in the second, and 5 in the third) is **2870410560**.