In a school there are $20$ teachers who teach mathematics or physics. Of these, $12$ teach mathematics and $4$ teach both physics and mathematics. How many teach physics ?
Let $M$ denote the set of teachers who teach mathematics and $P$ denote the set of teachers who teach physics. In the statement of the problem, the word 'or' gives us a clue of union and the word 'and' gives us a clue of intersection. We, therefore, have
$n( M \cup P )=20, n( M )=12 \text { and } n( M \cap P )=4$
We wish to determine $n( P ).$
Using the result $n( M \cup P )=n( M )+n( P )-n( M \cap P )$
we obtain $20=12+n(P)-4$
Thus $n( P )=12$
Hence $12$ teachers teach physics.
In a survey it was found that $21$ people liked product $A, 26$ liked product $B$ and $29$ liked product $C.$ If $14$ people liked products $A$ and $B, 12$ people liked products $C$ and $A, 14$ people liked products $B$ and $C$ and $8$ liked all the three products. Find how many liked product $C$ only.
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In a class of $100$ students, $55$ students have passed in Mathematics and $67$ students have passed in Physics. Then the number of students who have passed in Physics only is
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