Out of all the patients in a hospital $89 %$ are found to be suffering from heart ailment and $98 %$ are suffering from lungs infection. If $k %$ of them are suffering from both, then k can not belong to the set is

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$\{80, 83, 86, 89\}$

$\{84, 87, 90, 93\}$

$\{79, 81, 83, 85\}$

$\{84, 86, 88, 90\}$

answer is C.

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Detailed Solution

$n (A) = 89 %$ , $n (B) = 98 %$

$n (A \cup B) \leq 100 \Rightarrow n (A) + n (B) - n (A \cap B) \leq 100$

Now

$100 \geq 89 + 98 - n (A \cap B) \Rightarrow 87 \leq n (A \cap B)$

Now

$n (A \cap B) = \min \{n (A), n (B)\} \leq 89$

$\Rightarrow 87 \leq n (A \cap B) \leq 89$

$∴ n (A \cap B) \notin \{79, 81, 83, 85\}$

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