In how many ways can 100 soldiers be divided into 4 squads of 10, 20, 30, 40 respectively?

Options

A. 1700
B. 18!
C. 190
D. none of these

Answer

To solve the problem of dividing 100 soldiers into 4 squads of sizes 10, 20, 30, and 40 respectively, we can follow these steps:

### Step 1: Understand the problem
We need to divide 100 soldiers into 4 distinct groups with specified sizes: one group of 10 soldiers, one group of 20 soldiers, one group of 30 soldiers, and one group of 40 soldiers.

### Step 2: Calculate the number of ways to choose the first group
To form the first squad of 10 soldiers from 100 soldiers, we can choose 10 soldiers in \( \binom{100}{10} \) ways.

### Step 3: Calculate the number of ways to choose the second group
After selecting the first group, we have 90 soldiers left. We now need to select 20 soldiers for the second squad. This can be done in \( \binom{90}{20} \) ways.

### Step 4: Calculate the number of ways to choose the third group
Next, we have 70 soldiers remaining. We need to select 30 soldiers for the third squad, which can be done in \( \binom{70}{30} \) ways.

### Step 5: Calculate the number of ways to choose the fourth group
Finally, we have 40 soldiers left, and we need to select all of them for the fourth squad. This can be done in \( \binom{40}{40} \) ways, which is equal to 1.

### Step 6: Combine the selections
The total number of ways to form the squads is the product of the number of ways to choose each group:
\[ \text{Total Ways} = \binom{100}{10} \times \binom{90}{20} \times \binom{70}{30} \times \binom{40}{40} \]

### Step 7: Account for the indistinguishability of the squads
Since the squads are distinct (i.e., they have different sizes), we do not need to divide by any factorial.

### Final Answer
Thus, the total number of ways to divide the 100 soldiers into the specified squads is:
\[ \text{Total Ways} = \binom{100}{10} \times \binom{90}{20} \times \binom{70}{30} \times 1 \]