To find the total number of students who passed the examination, we need to first determine the total number of students who failed in at least one subject (either Kannada or Mathematics or both). We can then subtract this number from the total number of students who appeared for the examination.

Examining the Given Information

Let's list down the details provided in the question:

• Total number of students who appeared for the examination = $130$
• Number of students who failed in Kannada ($N(K)$) = $52$
• Number of students who failed in Mathematics ($N(M)$) = $72$
• Number of students who failed in both Kannada and Mathematics ($N(K \cap M)$) = $34$

Calculating Students Failed in At Least One Subject

We use the principle of inclusion-exclusion to find the total number of students who failed in at least one subject. This formula helps us to correctly count students without double-counting those who failed in both subjects.

The formula is: Number of students failed in at least one subject = (Number failed in Kannada) + (Number failed in Mathematics) - (Number failed in both Kannada and Mathematics)

In set theory notation, this is represented as:

$$N(K \cup M) = N(K) + N(M) - N(K \cap M)$$

Now, let's substitute the given values into the formula:

$$N(K \cup M) = 52 + 72 - 34$$

First, add the number of students who failed in each subject individually:

$$52 + 72 = 124$$

Next, subtract the number of students who failed in both subjects. This step is crucial because students who failed in both subjects were counted twice (once in Kannada failures and once in Mathematics failures):

$$124 - 34 = 90$$

So, the total number of students who failed in at least one subject (Kannada, Mathematics, or both) is $90$.

Determining the Total Number of Students Who Passed

To find the number of students who passed the examination, we subtract the total number of students who failed in at least one subject from the total number of students who appeared for the examination.

Total students passed = Total students appeared - Total students failed in at least one subject

Using the values we have:

Total students passed = $130 - 90$

Total students passed = $40$

Therefore, the total number of students who passed the examination is $40$.