Out of 130 students appearing in an examination, 62 failed in English, 52 failed in Mathematics, whereas 24 failed in both English and Mathematics. The number of students who passed finally is

The correct option is 40.

We can solve this problem step-by-step using set theory and a Venn diagram approach.

Let us define the sets for the students who failed in the examinations:
Let $E$ be the set of students who failed in English.
Let $M$ be the set of students who failed in Mathematics.
Let $U$ be the universal set representing the total number of students who appeared in the examination.

From the given data, we have the following values:
Total number of students, $n\left(U\right)=130$
Number of students who failed in English, $n\left(E\right)=62$
Number of students who failed in Mathematics, $n\left(M\right)=52$
Number of students who failed in both subjects, $n(E\cap M)=24$

To find the total number of students who failed in English, Mathematics, or both, we use the set union formula:
$n(E\cup M)=n\left(E\right)+n\left(M\right)-n(E\cap M)$

Substituting the given values into the formula:
$n(E\cup M)=62+52-24$
$n(E\cup M)=114-24$
$n(E\cup M)=90$

This means that out of the 130 students, 90 students failed in at least one of the two subjects (either English, Mathematics, or both).

The students who passed finally are those who did not fail in any of the subjects. Therefore, the number of students who passed is the total number of students minus the number of students who failed in at least one subject:
Number of passed students = $n\left(U\right)-n(E\cup M)$
Number of passed students = $130-90=40$

Thus, the number of students who passed finally is 40.