Question

In a car with seating capacity of exactly five persons, two persons can occupy the front seat and three persons can occupy the back seat. If amongst the seven persons, who wish to travel by this car, only two of them know driving, then the number of ways in which the car can be fully occupied and driven by them, is:

• A. 360
• B. 60
• C. 240
• D. 720

Hint:

When arranging people in specific seats, first choose the driver (if applicable), then select and arrange the remaining passengers in the available seats. Don't forget to use combinations and permutations as needed.

Answer 1

The Correct Option is A

Step 1: Choose the driver.
Out of the 7 people, only 2 can drive, so we have 2 choices for the driver.
Step 2: Choose the remaining passengers.
After selecting the driver, we need to select 4 more people out of the remaining 6. The number of ways to choose 4 people from 6 is given by: [ \binom64 = \frac6 × 52 × 1 = 15 ]
Step 3: Arrange the passengers.
After selecting the 4 passengers, we need to arrange them in the car. Since there are 2 seats in the front and 3 in the back, we first arrange the 2 chosen passengers in the front seats, which can be done in: [ P(2, 2) = 2! = 2 ] Then, we arrange the remaining 3 passengers in the back seats, which can be done in: [ P(3, 3) = 3! = 6 ]
Step 4: Total number of arrangements.
The total number of ways to arrange the passengers and the driver is: [ 2 × 15 × 2 × 6 = 360 ] Final Answer: 360.