In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and

Question

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m+n is equal to _______. [2024]

Smart Achievers · Accepted Answer

(45)

$Total number of students = n (S) = 220$

$n (M) \in [125, 130], n (P) \in [85, 95], n (C) \in [75, 90]$

$n (M \cup P \cup C) = 220 - 10 = 210$

$n (M \cap P) = 40, n (P \cap C) = 30, n (M \cap C) = 50$

$n (M \cup P \cup C) = \sum n (M) - \sum n (M \cap P) + n (M \cap P \cap C)$

$\Rightarrow n (M \cap P \cap C) = 210 + (40 + 30 + 50) - \sum n (M)$

$∵$ $(n {(M \cap P \cap C))}_{m a x} = n = \min {n (M \cap P), n (P \cap C), n (M \cap C)} = 30$

$∵$ $(\sum n {(M))}_{m a x} = 130 + 95 + 90 = 315$

$\Rightarrow (n {(M \cap P \cap C))}_{m i n} = m = 330 - 315 = 15$

$\Rightarrow n + m = 45$

Solution

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