Question

Priya is organizing seating arrangements for a panel discussion in Mumbai. She has 6 panelists: 2 men & 4 women. They need to be seated in a row such that no two men sit together. In how many ways can this seating arrangement be made?

• A. 576
• B. 480
• C. 288
• D. 144

Hint:

For "no two items together" problems, always arrange the unrestricted items first to create gaps, then place the restricted items into those gaps.

Answer 1

The Correct Option is B

Step 1: Understanding the Question: 
We need to arrange 2 men and 4 women in a row such that the 2 men are never adjacent to each other.

Step 2: Key Formula or Approach:
This problem can be solved using the "Gap Method".
1. First, arrange the items without the restriction (the women).
2. Then, place the restricted items (the men) in the gaps created by the first arrangement.
The number of ways to arrange \(n\) distinct items is \(n!\).
The number of ways to arrange \(r\) distinct items in \(n\) available spots is \(^nP_r = \frac{n!}{(n-r)!}\).

Step 3: Detailed Explanation:
Part 1: Arrange the women
First, let's seat the 4 women in a row. Since there are no restrictions on them, the number of ways to arrange 4 distinct women is:
 

\[4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}\]

Part 2: Place the men in the gaps
Arranging the 4 women creates 5 possible gaps where the men can be seated so that they are not together: 
 

\(W_1  W_2 W_3 W_4 \)

We need to place the 2 men in these 5 gaps. We can choose 2 gaps out of 5 and arrange the 2 men in these chosen gaps. The number of ways to do this is given by the permutation formula \(^5P_2\).
 

\[^5P_2 = \frac{5!}{(5-2)!} = \frac{5!}{3!} = 5 \times 4 = 20 \text{ ways}\]

Part 3: Total number of arrangements
The total number of valid seating arrangements is the product of the number of ways to arrange the women and the number of ways to place the men.
 

\[\text{Total Ways} = (\text{Ways to arrange women}) \times (\text{Ways to place men})\]

 

\[\text{Total Ways} = 24 \times 20 = 480\]

Step 4: Final Answer:
There are 480 ways to make the seating arrangement such that no two men sit together.