IPMAT Indore 2019 (MCQ) - In a class of 65 students 40 like cricket, 25 like football and 20 like hockey. 10 students like both cricket and football, 8 students like football and hockey and 5 students like all three sports. If all the students like at least one sport, then the number of students who like both cricket and hockey is | PYQs + Solutions | AfterBoards
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IPMAT Indore 2019

Modern Math
>
Set Theory

Medium

In a class of 65 students 40 like cricket, 25 like football and 20 like hockey. 10 students like both cricket and football, 8 students like football and hockey and 5 students like all three sports. If all the students like at least one sport, then the number of students who like both cricket and hockey is

✅ Correct Option: 1
We need to find n(C∩H)n(C \cap H), the number of students who like both cricket and hockey.
Using the set theory formula for three sets:
n(C∪F∪H)=n(C)+n(F)+n(H)−n(C∩F)−n(C∩H)−n(F∩H)+n(C∩F∩H)n(C \cup F \cup H) = n(C) + n(F) + n(H) - n(C \cap F) - n(C \cap H) - n(F \cap H) + n(C \cap F \cap H)
Substituting the given values:
65=40+25+20−10−n(C∩H)−8+565 = 40 + 25 + 20 - 10 - n(C \cap H) - 8 + 5
65=85−10−n(C∩H)−8+565 = 85 - 10 - n(C \cap H) - 8 + 5
65=72−n(C∩H)65 = 72 - n(C \cap H)
n(C∩H)=72−65=7n(C \cap H) = 72 - 65 = 7
Therefore, the number of students who like both cricket and hockey is 7.
Apart from the formula (which we highly recommend you learn), you can use this visual method to solve it too:

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