Question

The number of ways in which 3 men and 3 women can be arranged in a row of 6 seats, such that the first and last seats must be filled by men is:

• A. 720
• B. 36
• C. 144
• D. 72

Hint:

When arranging objects with restrictions, start by placing the restricted objects first and then arrange the remaining objects in the available positions.

Answer 1

The Correct Option is C

We are given that we have 3 men and 3 women, and we need to arrange them in a row of 6 seats. The first and last seats must be filled by men. 
Step 1: The first and last seats must be occupied by men. Since we have 3 men, we can choose a man for the first seat in 3 ways, and then we can choose a man for the last seat in 2 ways (since one man has already been seated). Therefore, the number of ways to arrange men in the first and last seats is: \[ 3 \times 2 = 6. \] 
Step 2: After placing the men in the first and last seats, we have 4 seats remaining. These 4 seats must be filled by the remaining 1 man and 3 women. The number of ways to arrange these 4 people in the remaining 4 seats is: \[ 4! = 24. \] 
Step 3: Thus, the total number of ways to arrange the 3 men and 3 women, with the first and last seats occupied by men, is: \[ 6 \times 24 = 144. \]