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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 $\mathrm{m}$ and $\mathrm{n}$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to .............................
$50$
$45$
$78$
$49$
Solution
$ 125 \leq \mathrm{m}+90-\mathrm{x} \leq 130 $
$ 85 \leq \mathrm{P}+70-\mathrm{x} \leq 95 $
$ 75 \leq \mathrm{C}+80-\mathrm{x} \leq 90 $
$ \mathrm{~m}+\mathrm{P}+\mathrm{C}+120-2 \mathrm{x}=210 $
$ \Rightarrow 15 \leq \mathrm{x} \leq 45 \& 30-\mathrm{x} \geq 0 $
$ \Rightarrow 15 \leq \mathrm{x} \leq 30 $
$ 30+15=45$
