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If 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 . The number of students who have opted Mathematics alone is
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Verified Answer
The correct answer is:
60
Let $M, P$ and $C$ be the set of students who opted Mathematics, Physics and Chemistry respectively.
Then, $n(M)=100, n(P)=70, n(C)=40$
$n(M \cap P)=30, n(M \cap C)=28, n(P \cap C)=23$,
$n(M \cap P \cap C)=18$
Number of students who opted Mathematics alone
$$
\begin{aligned}
&=n\left(M \cap P^{\prime} \cap C^{\prime}\right) \\
&=n\left(M \cap(P \cup C)^{\prime}\right) \\
&=n(M)-n(M \cap(P \cup C)) \\
&=n(M)-n((M \cap P) \cup(M \cap C)) \\
&=n(M)-n(M \cap P)-n(M \cap C)+n(M \cap P \cap C) \\
&=100-30-28+18=60
\end{aligned}
$$
Then, $n(M)=100, n(P)=70, n(C)=40$
$n(M \cap P)=30, n(M \cap C)=28, n(P \cap C)=23$,
$n(M \cap P \cap C)=18$
Number of students who opted Mathematics alone
$$
\begin{aligned}
&=n\left(M \cap P^{\prime} \cap C^{\prime}\right) \\
&=n\left(M \cap(P \cup C)^{\prime}\right) \\
&=n(M)-n(M \cap(P \cup C)) \\
&=n(M)-n((M \cap P) \cup(M \cap C)) \\
&=n(M)-n(M \cap P)-n(M \cap C)+n(M \cap P \cap C) \\
&=100-30-28+18=60
\end{aligned}
$$
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