In a survey of $600$ students in a school, $150$ students were found to be taking tea and $225$ taking coffee, $100$ were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

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Let $U$ be the set of all students who took part in the survey.

Let $T$ be the set of students taking tea.

Let $C$ be the set of students taking coffee.

Accordingly, $n(U)=600, n(T)=150, n(C)=225, n(T \cap C)=100$

To find : Number of student taking neither tea nor coffee i.e., we have to find $n\left(T^{\prime} \cap C^{\prime}\right)$

$n\left(T^{\prime} \cap C^{\prime}\right)=n(T \cup C)^{\prime}$

$=n(U)-n(T \cup C)$

$=n(U)-[n(T)+n(C)-n(T \cap C)]$

$=600-[150+225-100]$

$=600-275$

$=325$

Hence, $325$ students were taking neither tea nor coffee.

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