GATE IT 2004 | Question: 2

Question

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?

• A. $15$
• B. $20$
• C. $25$
• D. $30$

Comments

I think the question is wrong "35 students have taken both Data Structures and Computer Organization; 30 students have taken both Data Structures and Computer Organization"

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This is the standard Inclusion-exclusion Principle

Image source : (chatgpt)

 

Let the three courses be:

• $P$: Programming Language

• $D$: Data Structures

• $C$: Computer Organization

Given data in Question 

• Individual Course Counts:

  $$|P| = 125, \quad |D| = 85, \quad |C| = 65$$

• Pairwise Overlaps:

  $$|P \cap D| = 50, \quad |P \cap C| = 35, \quad |D \cap C| = 30$$

• Triple Overlap (All Three Courses):

  $$|P \cap D \cap C| = 15$$

• Total Universe (Total Students):

  $$\text{Total} = 200$$

Apply the Principle of Inclusion–Exclusion

The number of students who took at least one course is given by the formula:

$$|P \cup D \cup C| = |P| + |D| + |C| - |P \cap D| - |P \cap C| - |D \cap C| + |P \cap D \cap C|$$

Substitute the given values into the formula:

$$|P \cup D \cup C| = 125 + 85 + 65 - 50 - 35 - 30 + 15$$

$$|P \cup D \cup C| = 275 - 115 + 15$$

$$|P \cup D \cup C| = 175$$

 Students Who Took None

$$\text{Students who took none} = 200 - 175 = 25$$

Therefore, the correct option is C. 25.

$$\boxed{25}$$

Answer 1

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.

Comments

venn diagram will also do.
(may be proof of inclusion-exclusion follows from venn diagram, Not sure.)

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yes sir, it can be proved from Venn diagram.

Inclusion- exclusion if and only if Venn diagram.

anyone can be proved from one another.

Answer 2

 

•  A=Students who have taken Programming.
• B= Students who have taken Data Structures.
• C= Students who have taken Computer Organisation.

P(AUBUC)=P(A)+P(B)+P(C)- P(A∩B)-P(A∩C)- P(B∩C)+P(A∩B∩C)

125+85+65-50−35−30+15 =175

No of students not taking any courses-> 200−175=25

So C is the Answer.

😊😊😊😊😊😊😊😊😊😊😊😊

Answer 3

If we consider "35 students have taken both Programming language and Computer Organization; 30 students have taken both Data Structures and Computer Organization" then, correct ans is 25.

Option (C)25, is the correct answer.

Comments

This is beyond science!!!!

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haha!

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i am not getting what he is trying to say can u pls explain? @Thadymademe