An investigator interviewed 100...

MathematicsTypes of Sets and Set OperationsJEE Advanced 1978Easy

An investigator interviewed 100 students to determine their preferences for the three drinks : milk ( $M$ ), coffee ( $C$ ) and tea ( $T$ ). He reported the following : 10 students had all the three drinks $M, C$ and $T$ ; 20 had $M$ and $C$ ; 30 had $C$ and $T$ ; 25 had $M$ and $T$ ; 12 had $M$ only; 5 had $C$ only; and 8 had $T$ only. Using a Venn diagram find how many did not take any of the three drinks.

Answer:20

Visualized Solution (English)

Total Students .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n (U) = 100

Total students interviewed: $n (U) = 100$
Let $M$ , $C$ , and $T$ represent the sets of students who prefer Milk, Coffee, and Tea respectively.

The Triple Intersection .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n (M \cap C \cap T)

Given: $n (M \cap C \cap T) = 10$
This value is placed in the central region where all three circles overlap.

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n (M \cap C only)

Given: $n (M \cap C) = 20$
Students who take only Milk and Coffee: $n (M \cap C \cap T^{c}) = 20 - 10 = 10$

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n (C \cap T only)

Given: $n (C \cap T) = 30$
Students who take only Coffee and Tea: $n (C \cap T \cap M^{c}) = 30 - 10 = 20$

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n (M \cap T only)

Given: $n (M \cap T) = 25$
Students who take only Milk and Tea: $n (M \cap T \cap C^{c}) = 25 - 10 = 15$

Identifying 'Only' Regions

Given directly in the problem:
Students who take Milk only: $12$
Students who take Coffee only: $5$
Students who take Tea only: $8$

Union of Sets .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n (M \cup C \cup T)

Total students taking at least one drink is the sum of all regions inside the circles:
$n (M \cup C \cup T) = 12 + 5 + 8 + 10 + 20 + 15 + 10$
Summing these up: $n (M \cup C \cup T) = 80$

Students Taking No Drinks

Total students who did not take any drink:
$n (M \cup C \cup T)^{c} = n (U) - n (M \cup C \cup T)$
$100 - 80 = 20$
Final Answer: $20$ students did not take any of the three drinks.

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Visualized Solution (English)

Total Students $n (U) = 100$

The Triple Intersection $n (M \cap C \cap T)$

Calculating $n (M \cap C only)$

Calculating $n (C \cap T only)$

Calculating $n (M \cap T only)$

Identifying 'Only' Regions

Union of Sets $n (M \cup C \cup T)$

Students Taking No Drinks

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Visualized Solution (English)

Total Students .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(U)=100

The Triple Intersection .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(M∩C∩T)

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(M∩C only)

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(C∩T only)

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(M∩T only)

Identifying 'Only' Regions

Union of Sets .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(M∪C∪T)

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Total Students $n (U) = 100$

The Triple Intersection $n (M \cap C \cap T)$

Calculating $n (M \cap C only)$

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