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 1: Calculate the total arrangements without restrictions
The total number of ways to arrange \( n \) distinct objects is given by \( n! \) (n factorial).

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

### Step 2: Calculate the arrangements where the two particular objects are together
To count the arrangements where the two particular objects are together, we can treat these two objects as a single unit or block. This means we now have \( n - 1 \) units to arrange (the block of two objects plus the remaining \( n - 2 \) objects).

The number of ways to arrange these \( n - 1 \) units is \( (n - 1)! \).

**Hint:** When treating two objects as one, reduce the total count of objects by one.

### Step 3: Arrange the two particular objects within their block
Within the block of the two particular objects, these two objects can be arranged in \( 2! \) ways (since there are 2 objects).

**Hint:** The factorial of a number gives the number of ways to arrange that many distinct items.

### Step 4: Combine the arrangements
The total number of arrangements where the two particular objects are together is given by:
\[ (n - 1)! \times 2! \]

### Step 5: Calculate the arrangements where the two particular objects are never together
To find the arrangements where the two particular objects are never together, we subtract the arrangements where they are together from the total arrangements:
\[ \text{Arrangements where they are never together} = n! - (n - 1)! \times 2! \]

### Step 6: Simplify the expression
Substituting \( 2! = 2 \):
\[ \text{Arrangements where they are never together} = n! - 2(n - 1)! \]

This is the final expression for the number of ways to arrange \( n \) distinct objects in a line such that two particular objects are never together.

**Final Answer:**
\[ n! - 2(n - 1)! \]