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 Data Structures 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?

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 Data Structures 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?

(A) 15 

(B) 20 

(C) 25 

(D) 35 

(GATE 2004) 

Answer: (C) 25 

Explanation: 

Let $P$, $D$ and $C$ be the sets of students taking programming language, data structure and computer organization courses, respectively. 

Here, $n(P) = 125$, $n(D) = 85$, $n(C) = 65$, $n(P \cap D) = 50$, $n(P \cap C) = 35$, $n(D \cap C) = 30$ and $n(P \cap D \cap C) = 15$ 

[$n(X)$ denotes the number of elements present in the set $X$.] 

Now, $n(P \cup D \cup C)$ 

$= n(P) + n(D) + n(C) - n(P \cap D) - n(P \cap C) - n(D \cap C) + n(P \cap D \cap C)$ 

$= 125 + 85 + 65 - 50 - 35 - 30 + 15$ 

$= 175$ 

Therefore, the number of students taken no courses 

$= n(P^c \cap D^c \cap C^c)$ 

$= n(S) - n(P \cup D \cup C)$, where $S$ is the set of all students 

$= 200 - 175$ 

$= 25$