The number of ways of arranging 20 boys so that 3 particular boys are separated is:

Options

A. 9(16!)
B. 15(16!)
C. 15(17!)/2
D. none

Answer

To solve the problem of arranging 20 boys such that 3 particular boys are separated, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Boys**:
We have a total of 20 boys.

2. **Separate the Particular Boys**:
We need to ensure that 3 specific boys (let's call them A, B, and C) are not next to each other. This means we will first arrange the remaining boys.

3. **Arrange the Remaining Boys**:
If we exclude the 3 particular boys, we have 20 - 3 = 17 boys left. The number of ways to arrange these 17 boys is given by:
\[ 17! \]

4. **Calculate the Spaces Available**:
When we arrange 17 boys, there will be spaces before, between, and after them. For 17 boys, the number of spaces available is:
\[ 17 + 1 = 18 \text{ spaces} \]
(One space before the first boy, one space after each of the 16 boys in between, and one space after the last boy).

5. **Choose Spaces for the Particular Boys**:
We need to choose 3 out of these 18 spaces to place our particular boys A, B, and C. The number of ways to choose 3 spaces from 18 is given by the combination:
\[ \binom{18}{3} \]

6. **Arrange the Particular Boys**:
The 3 boys A, B, and C can be arranged among themselves in the chosen spaces. The number of ways to arrange these 3 boys is:
\[ 3! \]

7. **Combine the Results**:
The total number of arrangements of the 20 boys such that the 3 particular boys are separated is given by multiplying the arrangements of the 17 boys, the combinations of spaces, and the arrangements of the 3 boys:
\[ \text{Total Ways} = 17! \times \binom{18}{3} \times 3! \]

8. **Simplify the Expression**:
We can simplify \(\binom{18}{3}\) as follows:
\[ \binom{18}{3} = \frac{18!}{3! \times 15!} \]
Thus, the total ways can be rewritten as:
\[ 17! \times \frac{18!}{3! \times 15!} \times 3! \]
The \(3!\) cancels out:
\[ = 17! \times \frac{18!}{15!} \]
This simplifies further to:
\[ = 17! \times (18 \times 17 \times 16) \]
Therefore, the final expression becomes:
\[ = 18! \times 16 \times 17 \]

### Final Answer:
The total number of ways to arrange 20 boys so that 3 particular boys are separated is:
\[ 18! \times 16 \times 17 \]