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1.Set Theory
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In a class of $30$ pupils, $12$ take needle work, $16$ take physics and $18$ take history. If all the $30$ students take at least one subject and no one takes all three then the number of pupils taking $2$ subjects is
A
$16$
B
$6$
C
$8$
D
$20$
Solution
(a) Given $n(N) = 12$, $n(P) = 16$, $n(H) = 18$, $n(N \cup P \cup H) = 30$
From, $n(N \cup P \cup H) = n(N) + n(P) + n(H) – n(N \cap P)$
$ – n(P \cap H) – n(N \cap H) + n(N \cap P \cap H)$
$\therefore n(N \cap P) + n(P \cap H) + n(N \cap H) = 16$
Now, number of pupils taking two subjects
$ = n(N \cap P) + n(P \cap H) + n(N \cap H) – 3n(N \cap P \cap H)$
$ = 16 – 0 = 16$.
Standard 11
Mathematics
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