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 car can be fully occupied and driven by them.
A.360
B.60
C.240
D.720

Answer 1

Hint: This is a question on the concept of permutation and combination. Permutation is the arrangement of the object and combination is the selection of objects. we can find several ways to arrange and select the objects. In this case we have been given that the car has the seating capacity of 5 persons which means there are 5 objects to be arranged and selected.

Complete step-by-step answer:
Though initially we have 5 objects as a person but digging further into the question we observe 7 objects out of which we are restricted with the number of objects since only two of them know driving. Therefore there are 7 persons out of which 5 are drivers and 2 are non drivers.
Know selecting one driver out of the two which can occupy the front seat, Therefore the number of selections that can be formed from two different objects taken one at a time is ${{(^n}{C_r})^2}{C_1} = \dfrac{{\left| \!{\underline {\,
  2 \,}} \right. }}{{\left| \!{\underline {\,
  1 \,}} \right. \left| \!{\underline {\,
  {(2 - 1)} \,}} \right. }} = 2$
 The left four seats can be occupied by arranging five non drivers in the four leftover seats ,
$^5{P_4} = \dfrac{{\left| \!{\underline {\,
  5 \,}} \right. }}{{\left| \!{\underline {\,
  {(5 - 4)} \,}} \right. }} = \left| \!{\underline {\,
  5 \,}} \right. = 120$
Now only three more seats are left to be occupied so that the car is filled to its capacity and can be driven therefore the five persons can be arranged as
$^5{P_3} = \dfrac{{\left| \!{\underline {\,
  5 \,}} \right. }}{{\left| \!{\underline {\,
  {(5 - 3)} \,}} \right. }} = \dfrac{{\left| \!{\underline {\,
  5 \,}} \right. }}{{\left| \!{\underline {\,
  2 \,}} \right. }} = \dfrac{{120}}{2} = 60$
The total number of combination will therefore be $^2{C_1}{(^5}{P_4}{ + ^5}{P_3}) = 2(120 + 60) = 2 \times 180 = 360$
So, the correct answer is “Option A”.

Note: -The number of permutations of n distinct objects taken r at a time is the total number of arrangements of n distinct objects in groups of r where the order of arrangement is important. Arrangements can differ depending upon the conditions of whether the objects can be repeated or not.