Question

In a group of 30 students 8 take Hindi , 12 take Sanskrit and 3 take both languages. How many students of the group take neither Hindi nor Sanskrit?

Answer 1

Hint: We will find the number of students of the group who have at least one subject and then subtract the obtained value from the total number of students to get the number of students of the group who take neither Hindi nor Sanskrit.

Complete answer:
Total number of students are 30 and it is given that 8 students take Hindi and 12 students take Sanskrit and 3 take both languages.
We have to find the number of students in the group who take neither of the languages.
First, we will find the number of students who take at least one language, sw we will apply the formula here which is,
$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$
Here, $P\left( {A \cup B} \right)$ represents the number of students who take at least one subject.
$P\left( A \right)$ represents the students who take Hindi.
$P\left( B \right)$ represents the students who take Sanskrit.
$P\left( {A \cap B} \right)$ represents the students who take both the language Hindi and Sanskrit.
So we have,
$
  P\left( A \right) = 8 \\
  P\left( B \right) = 12 \\
  P\left( {A \cap B} \right) = 3 \\
$
No, we will put the values in the formula $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$
$
  P\left( {A \cup B} \right) = 8 + 12 + 3 \\
  P\left( {A \cup B} \right) = 20 - 3 \\
  P\left( {A \cup B} \right) = 17 \\
$
So, the number of students who take at least one language is 17.
Now, we will subtract the values of the number of students who take at least one language from the value of the total number of students to get the value of the students of the group who take neither of the subject.
So, the required value $ = 30 - 17 = 13$
So, there are 13 students in a group of 30 students who take neither of the subjects.

Note: Here $P\left( {A \cup B} \right)$ does not represent total number of students but represents the total number of students who take at least one subject out of two subjects. So we found $P\left( {A \cup B} \right)$ first and then subtract it from 30 to get the required value.