In how many ways can 10 soldiers stand in two rows having 5 soldiers in each row?

Options

A. 3628800
B. 7257600
C. 35400
D. 985000

Answer

To solve the problem of how many ways 10 soldiers can stand in two rows with 5 soldiers in each row, we can follow these steps:

### Step 1: Choose 5 soldiers for the first row
We need to select 5 soldiers from a total of 10 soldiers. The number of ways to choose 5 soldiers from 10 is given by the combination formula:

\[ \binom{10}{5} \]

### Step 2: Arrange the 5 soldiers in the first row
Once we have chosen 5 soldiers for the first row, we can arrange these 5 soldiers in that row. The number of arrangements of 5 soldiers is given by:

\[ 5! \]

### Step 3: Choose 5 soldiers for the second row
After selecting 5 soldiers for the first row, we have 5 soldiers left. We need to place all of these remaining soldiers in the second row. The number of ways to choose these 5 soldiers from the remaining 5 is:

\[ \binom{5}{5} = 1 \]

### Step 4: Arrange the 5 soldiers in the second row
Similar to the first row, we can arrange these 5 soldiers in the second row. The number of arrangements is again:

\[ 5! \]

### Step 5: Consider the interchange of rows
Since the two rows are distinct, we can also interchange the rows. This gives us a factor of 2 for the arrangements.

### Final Calculation
Now, we can combine all these calculations to find the total number of ways:

\[ \text{Total Ways} = \binom{10}{5} \times 5! \times \binom{5}{5} \times 5! \times 2 \]

Substituting the values:

\[ = \binom{10}{5} \times 5! \times 1 \times 5! \times 2 \]

Calculating \(\binom{10}{5}\):

\[ \binom{10}{5} = \frac{10!}{5! \times (10-5)!} = \frac{10!}{5! \times 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]

Now substituting back:

\[ = 252 \times 5! \times 5! \times 2 \]

Calculating \(5!\):

\[ 5! = 120 \]

Thus,

\[ = 252 \times 120 \times 120 \times 2 \]

Calculating:

\[ = 252 \times 14400 = 3628800 \]

Thus, the total number of ways 10 soldiers can stand in two rows of 5 soldiers each is:

\[ \text{Total Ways} = 3628800 \]

### Final Answer
The total number of ways in which 10 soldiers can stand in two rows having 5 soldiers in each row is **3628800**.

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