Question

In a car agency one day 120 cars were decorated with three different accessories viz., power window, AC and, music system. 80 cars were decorated with power windows, 84 cars were decorated with AC and 80 cars were decorated with music systems. What is the minimum and maximum number of cars which were decorated with all of three accessories?

• A. 10, 61
• B. 10, 45
• C. 25, 35
• D. None of these

Answer 1

The correct option is D None of these

Given, sum of all cars representing all the accessories = S = I + 2 II + 3 III = 80 + 84 + 80 = 244 - - - (i)
Number of cars having at least one accessory = X = I + II + III = 120 - - - (ii)
Subtracting ii) from i), we get
II + 2 III = 124 - - - (iii)
2 III = 124 - II
III = $62-\frac{II}{2}$
III takes the maximum value when II = 0 $\Rightarrow$ $II{I}_{max}$ = 62
Also, subtracting ii) from iii), we get
III - I = 4 $\Rightarrow$ III = I + 4
For III to have minimum value, I = 0
$\therefore II{I}_{min}=4$

Alternatively:

80 cars were decorated with power windows it means at least 40 cars were decorated with AC or music system or both. 84 cars were decorated with ACs, it means at-least 36 cars were decorated with power windows or music systems or both. 80 cars were decorated with music system means at least 40 cars were decorated with power windows or ACs or both. It means if there is no intersection in these three, then at most 40 + 36 + 40 = 116 cars had been decorated with one or two accessories.
Hence at least 4 cars would have been decorated with all three accessories.
For maximum value of x:

Music System
Total number of cars = (80 - x) + (84 - x) + (80 - x) + x
120 = 244 - 2x
$\Rightarrow 2x=124\Rightarrow x=62$
minimum $\to$ 4 cars and maximum $\to$ 62 cars