Question

A total of 1232 students have taken a course in Tamil, 879 have taken a course in Telugu, 114 have taken a course Hindi. Further 103 have taken a course both in Tamil and Telugu, 23 have taken course in Tamil and Hindi and 14 have taken course in Telugu and Hindi. If 2092 students have taken at least one of the Tamil, Telugu and Hindi, how many students have taken the course in all the 3 languages?

Answer 1

Answer:

7 students have taken the course in Tamil, Telugu, and Hindi.

Step-by-step explanation:

Given:

A sum of 1232 students have accepted a course in Tamil, 879 have accepted a course in Telugu, and 114 have accepted a course in Hindi. Further 103 have accepted a course both in Tamil and Telugu, 23 have accepted courses in Tamil and Hindi and 14 have accepted a course in Telugu and Hindi.

To find:

How many students have brought the course in all the 3 languages.

Step 1

Let A, B, and C represent the number of students who have taken a course in Tamil, Telugu, and Hindi respectively. Then

$|A|=1232, |B|=879$ and $|C|=114$

$|A \cap B|=103,$ $|A \cap C|=23,$ $|B \cap C|=14,$ and $|A \cup B \cup C|=2092$

Step 2

By principles of Inclusion-Exclusion, we have

$|A \cup B \cup C| &=|A|+|B|+|C|-|A \cap B|-|A \cap C|-|B \cap C|+|A \cap B \cap C|$

$|A \cap B \cap C| &=|A \cup B \cup C|-|A|-|B|-|C|+|A \cap B|+|A \cap C|+|B \cap C|$

Substituting the values of $|A|=1232, |B|=879$ and $|C|=114$

$|A \cap B \cap C| &=2092-1232-879-114+103+23+14$

Simplifying the above equation as

$|A \cap B \cap C| &=2232-2225$

$|A \cap B \cap C| &=7\end{aligned}$

Therefore, 7 students have taken the course in Tamil, Telugu, and Hindi.

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