Sets, Relations & Functions
Sets — Survey / Inclusion-Exclusion (Three Sets)
nta_pyq_2024_apr
Grade 11
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 $\mathrm{m}$ 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 _____
Step-by-Step Solution
Key Concept: Let $x$ = number studying all three. By inclusion-exclusion: $|M|+|P|+|C| - (|M\cap P|+|P\cap C|+|C\cap M|) + x + 10 = 220$, so $|M|+|P|+|C| = 210 + 2x - 120 + x$... use the Venn diagram regions. With $|M\cap P|=40$, $|P\cap C|=30$, $|C\cap M|=50$: $M+P+C+120-2x=210 \Rightarrow M+P+C=90+2x$.
Using inclusion-exclusion with the Venn diagram and given constraints, $15\leq x\leq30$. $\mathrm{m}+\mathrm{n}=15+30=45$.
Correct Answer: 45