Answer these questions independently of each other.
In how many ways can 12 soldiers stand in two queue such that each queue has 6 soldiers?

Options

A. `^12P_6xx^12P_6xx2!`
B. `^12P_6xx^6P_6`
C. `6!xx6!xx2!`
D. `^12P_6xx6!xx2!`

Answer

To solve the problem of how many ways 12 soldiers can stand in two queues such that each queue has 6 soldiers, we can follow these steps:

### Step 1: Choose 6 soldiers for the first queue
We need to select 6 soldiers out of the 12 available. The number of ways to choose 6 soldiers from 12 is given by the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

For our case, it will be:

\[ \binom{12}{6} = \frac{12!}{6!6!} \]

### Step 2: Arrange the soldiers in the first queue
Once we have selected the 6 soldiers for the first queue, we can arrange these 6 soldiers in any order. The number of ways to arrange 6 soldiers is given by the factorial of 6:

\[ 6! \]

### Step 3: Arrange the soldiers in the second queue
The remaining 6 soldiers (who were not chosen for the first queue) will automatically go to the second queue. We can also arrange these 6 soldiers in any order, which is again:

\[ 6! \]

### Step 4: Consider the two queues as distinct
Since the two queues are distinct (Queue 1 and Queue 2), we need to multiply our result by 2! (the number of ways to arrange the two queues):

\[ 2! \]

### Step 5: Combine all the parts
Now, we can combine all the parts to find the total number of arrangements:

\[ \text{Total Ways} = \binom{12}{6} \times 6! \times 6! \times 2! \]

### Step 6: Calculate the values
Now we can compute the values:

1. Calculate \(\binom{12}{6}\):

\[ \binom{12}{6} = \frac{12!}{6!6!} = 924 \]

2. Calculate \(6!\):

\[ 6! = 720 \]

3. Calculate \(2!\):

\[ 2! = 2 \]

### Step 7: Substitute and calculate the total
Now we substitute these values into our total ways formula:

\[ \text{Total Ways} = 924 \times 720 \times 720 \times 2 \]

Calculating this gives:

\[ \text{Total Ways} = 924 \times 720^2 \times 2 = 924 \times 518400 = 479001600 \]

Thus, the total number of ways 12 soldiers can stand in two queues of 6 soldiers each is **479001600**.

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