In a class of 100 students, 12 students drink only milk and 5 students drink only coffee and 8 students drink only tea. Other report says 30 students take both coffee and tea, 25 students take milk and tea and 20 students take only milk and coffee. 10 students drink all the three. Find the number of students who do not drink anything.

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Solution

Given data:

Total number of students = 100

Number of students drink only milk = 12

Number of students drink only coffee = 5

Number of students drink only tea = 8

Number of students drink both coffee and tea n(B $\cup$ C)= 30

Number of students drink both milk and tea n(A $\cup$ C)= 25

Number of students drink both milk and coffee n(A $\cup$ B)= 20

Number of students drink all milk, coffee and tea n(A $\cap$ B $\cap$ C) = 10

Representing given data on the Venn-diagram

Number of students who do not drink anything = Total number of students - Number of students who drink at least one item

= 100 - Number of students who drink at least one item --------------------(1)

From the Venn-diagram

Number of students who drink at least one item = Number of students drink only milk +Number of students drink only coffee + Number of students drink only tea + Number of students drink both coffee and tea + Number of students drink both milk and tea + Number of students drink both milk and coffee - 2 $\times$ Number of students drink all milk, coffee and tea

{We subtracted the 2 $\times$ Number of students drink all milk, coffee and tea because we added it thrice while adding n(A $\cup$ B), n(B $\cup$ C), n(A $\cup$ C). So, we should subtract 2 $\times$ Number of students drink all milk, coffee and tea}

= 12 + 5 + 8 + 20 + 30 + 25 - 2 $\times$ 10

= 90

Number of students who do not drink anything = 100 - Number of students who drink at least one item

= 100 - 90 = 10

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