Question

# In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone? (a) 35 (b) 48 (c) 60 (d) 22 (e) 30

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Solution

## (c) 60 Let M, P and C denote the sets of students who have opted for mathematics, physics, and chemistry, respectively. Here, $n\left(M\right)$ = 100, $n\left(P\right)$ = 70 and $n\left(C\right)$ = 40 Now, $n\left(M\cap P\right)=30,n\left(M\cap C\right)=28,n\left(P\cap C\right)=23\mathrm{and}n\left(M\cap P\cap C\right)=18$ Number of students who opted for only mathematics: $n\left(M\cap P\text{'}\cap C\text{'}\right)=n\left\{M\cap \left(P\cup C\right)\text{'}\right\}\phantom{\rule{0ex}{0ex}}=n\left(M\right)-n\left\{M\cap \left(P\cup C\right)\right\}\phantom{\rule{0ex}{0ex}}=n\left(M\right)-n\left\{\left(M\cap P\right)\cup \left(M\cap C\right)\right\}\phantom{\rule{0ex}{0ex}}=n\left(M\right)-\left\{n\left(M\cap P\right)+n\left(M\cap C\right)-n\left(M\cap P\cap C\right)\right\}\phantom{\rule{0ex}{0ex}}=100-\left(30+28-18\right)\phantom{\rule{0ex}{0ex}}=60$ Therefore, the number of students who opted for mathematics alone is 60

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