Question

In an examination, out of $100$ students, $75$ passed in English, $60$ passed in Mathematics and $45$ passed in both English and Mathematics. The number of students who passed in none of the two subjects, is
(Assume all students gave both the exams)

• A. $5$
• B. $10$
• C. $20$
• D. $40$

Answer 1

The correct option is B $10$

Let $U$ be the universal set.
Let $E$ denote the set of students who passed in English,
and $M$ denote the set of students who passed in Mathematics.
$n\left(U\right)=100$
$n\left(E\right)=75,n\left(M\right)=60$ and $n(E\cap M)=45$
$n(E\cup M)=n\left(E\right)+n\left(M\right)-n(E\cap M)$
$=75+60-45$
$=90$

Now, $n(E^{\prime }\cap M^{\prime })=n(E\cup M{)}^{\prime }$
$=n\left(U\right)-n(E\cup M)$
$=100-90=10$