An investigator interviewed $100$ students to determine the performance of three drinks : milk, coffee and tea. The investigator reported that $10$ students take all three drink milk, coffee and tea ; $20$ students take milk and coffee; $25$ students take milk and tea ; $20$ students take coffee and tea; $12$ students take milk only; $5$ students take coffee only and $8$ students take tea only. Then the number of students who did not take any of three drinks is

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Solution

Correct option is D. $30$
Let,
$n (M)$ be the number of students who drink milk only

$n (T)$ be the number of students who drink tea only

$n (C)$ be the number of students who drink coffee only

Given:
$n (M) = 12, n (C) = 5$ and $n (T) = 8, n (M \cap C \cap T) = 10$

$\Rightarrow n (M \cap C) = 20 - n (M \cap C \cap T) = 20 - 10 = 10$

Also $n (M \cap T) = 25 - n (M \cap C \cap T) = 25 - 15$

Also $n (T \cap C) = 20 - n (M \cap C \cap T) = 20 - 10 = 10$

$∴$ Number of students who take all the drinks

$= n (M) + n (C) + n (T) + n (M \cap C) + n (M \cap T) + n (T \cap C) + n (M \cap C \cap T)$

$= 12 + 5 + 8 + 10 + 15 + 10 + 10 = 70$

$∴$ Number of students who did not take any of the drink $= 100 - 80 = 30$ students.

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