**Concept: The question tests on Overlapping sets.**

**Solution:**

What is the overall necessity of the question here?

It is to

**minimize the number of students**, who watch**at least one channel.**=>We have to

Total = Fox News + Sky Sports+ABC- (Two channel viewership)+ Three channel +None

=>250 = 120 + 80 + 90 - (50 + 60 + 65) + three channel + none

**maximize the number of the students who watch none of these channels.**Total = Fox News + Sky Sports+ABC- (Two channel viewership)+ Three channel +None

=>250 = 120 + 80 + 90 - (50 + 60 + 65) + three channel + none

Let the three channel viewership be x

=>250 = 290 - 175 + x + none

=>250 = 115 + x + none

=>135 - x = none

=>We need to minimize the value of x as we are trying to maximise “none”

=>250 = 290 - 175 + x + none

=>250 = 115 + x + none

=>135 - x = none

=>We need to minimize the value of x as we are trying to maximise “none”

=> Let the number of students who watch ABC only =a

=>Thus we have (65 – x) students watching ABC and Fox (but not Sky) and (60 – x) students watching ABC and Sky (but not Fox) and for all the students who watch ABC ,

=>(65 - x) + (60 - x) + x + a = 90

=>125 - x + a = 90

=>35 = x- a

Since we want to minimize the value of x, so a must be 0.

=>(65 - x) + (60 - x) + x + a = 90

=>125 - x + a = 90

=>35 = x- a

Since we want to minimize the value of x, so a must be 0.

=>Thus x = 35(all three channels)

=>None = 135 - x = 135 - 35 = 100.

=>There could be a maximum 100 students who watch none of the 3 channels,

=>None = 135 - x = 135 - 35 = 100.

=>There could be a maximum 100 students who watch none of the 3 channels,

=>There must be a minimum of 250 - 100 = 150 who watch at least one channel.