First slide

Combinations

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 number of ways in which the car can be fully occupied and driven by them, is:

Moderate

A

360
B

60
C

240
D

720

Solution

Two cases arise:
Case 1: Both the drivers are occupying the car. In this case we have to choose 3 persons out of 5 for the remaining seats and car can be driven by any of the two drivers. Thus, in this case car can be occupied and driven in $(^{5} C_{3}) (^{2} C_{1}) (4!) = 480$
Case 2: Exactly one drive is occupying the car. In this case we have to choose 4 persons out of the remaining 5 people. In this case car can be occupied and driven in $(^{5} C_{4}) (^{2} C_{1}) (4!) = 240$ ways
Thus, the required number of ways $= 480 + 240 = 720$

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