The number of ways of arranging `n(gt2)` distinct objects in a line so that two particulars objects are never together is

Options

A. `(n-2)!.^(n-1)P_(2)`
B. `2(n-1)!`
C. `n!-2(n-1)!`
D. None of these

Answer

To find the number of ways to arrange `n` distinct objects in a line such that two particular objects are never together, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Total Arrangements**:
The total number of ways to arrange `n` distinct objects is given by the factorial of `n`, which is:
\[ \text{Total arrangements} = n! \]

2. **Consider the Case Where Two Particular Objects Are Together**:
Let's denote the two particular objects as A and B. If we consider A and B as a single unit or block, we can treat them as one object. Therefore, we now have:
- 1 block (A and B together)
- \(n - 2\) other distinct objects

This gives us a total of \(n - 1\) objects to arrange (the block plus the other objects).

3. **Calculate Arrangements for the Block**:
The number of ways to arrange the \(n - 1\) objects (the block and the other objects) is:
\[ (n - 1)! \]
Since the block (A and B) can be arranged in 2 ways (AB or BA), we multiply by 2:
\[ \text{Arrangements with A and B together} = 2 \times (n - 1)! \]

4. **Calculate Arrangements Where A and B Are Never Together**:
To find the arrangements where A and B are never together, we subtract the arrangements where they are together from the total arrangements:
\[ \text{Arrangements where A and B are never together} = n! - 2 \times (n - 1)! \]

5. **Final Result**:
Thus, the number of ways to arrange `n` distinct objects such that two particular objects are never together is:
\[ n! - 2 \times (n - 1)! \]

### Conclusion:
The correct answer is:
\[ \text{Option 3: } n! - 2 \times (n - 1)! \]