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In a class of $200$ students, $125$ students have taken Programming Language course, $85$ students have taken Data Structures course, $65$ students have taken Computer Organization course; $50$ students have taken both Programming Language and Data Structures, $35$ students have taken both Programming Language and Computer Organization; $30$ students have taken both Data Structures and Computer Organization, $15$ students have taken all the three courses.

How many students have not taken any of the three courses?

- $15$
- $20$
- $25$
- $30$

41 votes

Best answer

The question has a slight misprint. It should be what Bhagirathi says in the comments.

Nevertheless,

$\small \Bigl | A \cup B \cup C \Bigr | = |A| + |B| + |C| - \Bigl | A \cap B \Bigr | - \Bigl | A \cap C \Bigr | - \Bigl | B \cap C \Bigr | + \Bigl | A \cap B \cap C \Bigr |$

- $A \equiv $ Students who have taken Programming.
- $B \equiv $ Students who have taken Data Structures.
- $C \equiv $ Students who have taken Computer Organisation.

So, the number of students who have taken any of the $3$ courses is given by:

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

$ \qquad\qquad\quad \;= 125 + 85 + 65 - 50 - 35 - 30 + 15= 175$

Therefore, the number of students who haven't taken any of the $3$ courses is: $200 - 175 = 25$

**Hence, the answer is Option C.**