In how many ways can n women be seated in a row so that a particular women will not be next to each other?

Options

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

Answer

To solve the problem of seating \( n \) women in a row such that a particular woman does not sit next to another specific woman, we can follow these steps:

### Step 1: Calculate the total arrangements without restrictions
The total number of ways to arrange \( n \) women in a row is given by the factorial of \( n \):
\[ \text{Total arrangements} = n! \]

**Hint:** Remember that the factorial of a number \( n \) represents the number of ways to arrange \( n \) distinct objects.

### Step 2: Calculate the arrangements where the two specific women are together
To find the arrangements where a particular woman (let's call her Woman A) is next to another specific woman (Woman B), we can treat these two women as a single entity or "block".

- When we consider Woman A and Woman B as one block, we effectively have \( n - 1 \) entities to arrange (the block plus the remaining \( n - 2 \) women).
- The number of ways to arrange these \( n - 1 \) entities is:
\[ (n - 1)! \]

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

### Step 3: Account for the arrangements within the block
Within the block of Woman A and Woman B, these two women can be arranged in 2 ways (either A next to B or B next to A). Therefore, the total arrangements where Woman A is next to Woman B is:
\[ 2 \times (n - 1)! \]

**Hint:** Always consider the internal arrangements when grouping items together.

### Step 4: Calculate the arrangements where Woman A is not next to Woman B
To find the arrangements where Woman A is not next to Woman B, we subtract the arrangements where they are together from the total arrangements:
\[ \text{Arrangements where A is not next to B} = n! - 2 \times (n - 1)! \]

**Hint:** Use subtraction to find the complement when dealing with restrictions.

### Step 5: Simplify the expression
Now, we can simplify the expression:
\[ n! - 2 \times (n - 1)! = n \times (n - 1)! - 2 \times (n - 1)! \]
Factoring out \( (n - 1)! \):
\[ = (n - 1)! \times (n - 2) \]

**Hint:** Factoring can help simplify complex expressions.

### Final Result
Thus, the number of ways to seat \( n \) women such that a particular woman does not sit next to another specific woman is:
\[ (n - 2) \times (n - 1)! \]

**Conclusion:** The correct answer is option A: \( n - 2 \times (n - 1)! \).