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 1: Arrange the 17 boys
First, we arrange the remaining 17 boys. The number of ways to arrange 17 boys is given by \(17!\).

**Hint:** Remember that the factorial notation \(n!\) represents the product of all positive integers up to \(n\).

### Step 2: Identify spaces created by the arrangement
When we arrange 17 boys, they create spaces around them. Specifically, there will be 18 spaces (1 space before the first boy, 1 space after each boy, and 1 space after the last boy).

**Hint:** If you have \(n\) items arranged in a line, the number of spaces created is \(n + 1\).

### Step 3: Place the 3 particular boys in the spaces
We need to place the 3 particular boys in these 18 spaces. To ensure that they are separated, we can choose any 3 out of the 18 spaces. The number of ways to choose 3 spaces from 18 is given by \(\binom{18}{3}\).

**Hint:** The binomial coefficient \(\binom{n}{r}\) gives the number of ways to choose \(r\) items from \(n\) items without regard to the order of selection.

### Step 4: Arrange the 3 particular boys
After selecting the 3 spaces, we can arrange the 3 particular boys in those spaces. The number of ways to arrange 3 boys is \(3!\).

**Hint:** The factorial \(3!\) is equal to \(3 \times 2 \times 1\).

### Step 5: Combine the arrangements
Now, we can combine the arrangements of the 17 boys, the selection of spaces, and the arrangement of the 3 boys. The total number of arrangements is given by:

\[ \text{Total arrangements} = 17! \times \binom{18}{3} \times 3! \]

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

1. Calculate \(17!\).
2. Calculate \(\binom{18}{3} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = 816\).
3. Calculate \(3! = 6\).
4. Finally, multiply these values together to get the total arrangements.

Putting it all together, we have:

\[ \text{Total arrangements} = 17! \times 816 \times 6 \]

### Final Answer
This gives us the total number of ways to arrange 20 boys such that 3 particular boys are separated.

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