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Question

# In a class there are 200 students, at least 140 of students like Maths, at least 150 like Science and at least 160 like English. All students like atleast 1 subject. What is the minimum number of students who like all three subjects?

A

50

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B

83

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C

100

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D

150

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Solution

## The correct option is A 50 We know that, a+d+g+f≥140 b+d+g+e≥150 c+e+f+g≥160 Adding these inequalities we get, (a+b+c)+2(d+e+f)+3g≥450 ⋯(i) We know that, a+b+c+d+e+f+g=200 ⋯(ii) Multiplying (ii) with 2 and subtracting it from (i) we get, g−(a+b+c)≥50 ⇒g≥50+(a+b+c) (a+b+c) is the number of students who like 1 subject alone. for g to be minimum, (a+b+c)=0 and hence, g≥50 So a minimum of 50 students like all 3 subjects

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