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  4. In a class of 30 pupils. 12 take needls 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

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Q. In a class of $30$ pupils. $12$ take needls 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

J & K CETJ & K CET 2005

Solution:

Given that $ n(N)=12,\,n(P)=16,\,n(H)=18, $
$ n(N\cup P\cup H)=30 $
and $ n(N\cap P\cap H)=0 $
Now, $ n(N\cup P\cup H)=n(N)+n(P)+n(H) $
$ -n(N\cup P)-n(P\cap H)-n(H\cap N) $
$ +n(N\cap P\cap H) $
$ \Rightarrow $ $ n(N\cap P)+N(P\cap H)+N(H\cap N) $
$ =n(N)+n(P)+n(H)-n(N\cup P\cup H) $
$ =(12+16+18)-30 $
$ =46-30=16 $