JEE MAIN - Mathematics (2024 - 30th January Morning Shift - No. 21)
A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics and Chemistry. It was found that all students passed in atleast one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, atmost 11 students passed in both Mathematics and Physics, atmost 15 students passed in both Physics and Chemistry, atmost 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _________.
Answer
10
Explanation
$\begin{aligned} & n(M)=20 \\\\ & n(P)=25 \\\\ & n(C)=16 \\\\ & n(M \cap P)=11 \\\\ & n(P \cap C)=15 \\\\ & n(M \cap C)=15\end{aligned}$
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$$ \begin{aligned} & n(C \cup P \cup M) \leq n(U)=40 . \\\\ & n(C)+n(P)+n(M)-n(C \cap M)-n(P \cap M)-n(C \cap \\\\ & P)+n(C \cap P \cap M) \leq 40 \\\\ & 20+25+16-11-15-15+x \leq 40 \\\\ & x \leq 20 \end{aligned} $$
But $11-x \geq 0$ and $15-x \geq 0$
$$ \Rightarrow x \geq 11 $$
$$ \begin{aligned} & n(C \cup P \cup M) \leq n(U)=40 . \\\\ & n(C)+n(P)+n(M)-n(C \cap M)-n(P \cap M)-n(C \cap \\\\ & P)+n(C \cap P \cap M) \leq 40 \\\\ & 20+25+16-11-15-15+x \leq 40 \\\\ & x \leq 20 \end{aligned} $$
But $11-x \geq 0$ and $15-x \geq 0$
$$ \Rightarrow x \geq 11 $$
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