JEE Main 2024 (Online) 4th April Morning ShiftQuestion 1
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 $\_\_\_\_\_\_\_\_$ .
Sets, Relations and Function
Answer$45$
Solution
$|M\cup P\cup C|=220-10=210=M+P+C-120+x$, so $x=330-(M+P+C)$ with $285\le M+P+C\le315$, and $x\le30$ (from $|P\cap C|=30$). All Venn regions stay non-negative for $15\le x\le30$, so $m=15$, $n=30$ and $m+n=45$.