A market research group conducted a survey of $1000$ consumers and reported that $720$ consumers like product $\mathrm{A}$ and $450$ consumers like product $\mathrm{B}$, what is the least number that must have liked both products?
Let $U$ be the set of consumers questioned, $S$ be the set of consumers who liked the product $A$ and $T$ be the set of consumers who like the product $B.$ Given that
$n( U )=1000, n( S )=720, n( T )=450$
So $ n( S \cup T ) =n( S )+n( T )-n( S \cap T ) $
$=720+450-n( S \cap T )=1170-n( S \cap T ) $
Therefore, $n( S \cup T )$ is maximum when $n( S \cap T )$ is least.
But $S \cup T \subset U$ implies $n( S \cup T ) \leq n( U )=1000 .$
So, maximum values of $n( S \cup T )$ is $1000 .$
Thus, the least value of $n( S \cap T )$ is $170 .$
Hence, the least number of consumers who liked both products is $170$
There are $200$ individuals with a skin disorder, $120$ had been exposed to the chemical $C _{1}, 50$ to chemical $C _{2},$ and $30$ to both the chemicals $C _{1}$ and $C _{2} .$ Find the number of individuals exposed to
Chemical $C_{2}$ but not chemical $C_{1}$
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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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A survey shows that $63 \%$ of the people in a city read newspaper $A$ whereas $76 \%$ read newspaper $B$. If $x \%$ of the people read both the newspapers, then a possible value of $x$ can be