In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most

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 $\_\_\_\_$ .\

Professor PC Thomas Classes · Answer

(45): Total number of students $=n(S) =220$\ $n(M) \in[125,130], n(P) \in[85,95]$,\ $n(C) \in[75,90]$\ $n(M \cup P \cup C)=220-10=210$\ $n(M \cap P)=40, n(P \cap C)=30$,\ $n(M \cap C)=50$\ $n(M \cup P \cup C)$\ $=\sum n(M)-\sum n(M \cap P)+n(M \cap P \cap C)$\ $\Rightarrow n(M \cap P \cap C)$\ $=210+(40+30+50)-\sum n(M)$\ $\because \quad(n(M \cap P \cap C))_{\max }=n$\ $=\min \{n(M \cap P)$,\ $n(P \cap C), n(M \cap C)\}=30$\ $\because \quad\left(\sum n(M)ight)_{\max }=130+95+90=315$\ $\Rightarrow \quad(n(M \cap P \cap C))_{\min }=m=330-315$\ $=15$\ $\Rightarrow n+m=45$\