# In a survey of 600 students in a school,

Question:

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?

Solution:

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, $n(\mathrm{U})=600, n(\mathrm{~T})=150, n(\mathrm{C})=225, n(\mathrm{~T} \cap \mathrm{C})=100$

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

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

$=n(\mathrm{U})-n(\mathrm{~T} \cup \mathrm{C})$

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

$=n(\mathrm{U})-[n(\mathrm{~T})+n(\mathrm{C})-n(\mathrm{~T} \cap \mathrm{C})]$

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

$=600-275$

$=325$

Hence, 325 students were taking neither tea nor coffee.