GATE CSE 2019 | Question: GA-9

Question

In a college, there are three student clubs, $60$ students are only in the Drama club, $80$ students are only in the Dance club, $30$ students are only in Maths club, $40$ students are in both Drama and Dance clubs, $12$ students are in both Dance and Maths clubs, $7$ students are in both Drama and Maths clubs, and $2$ students are in all clubs. If $75 \%$ of the students in the college are not in any of these clubs, then the total number of students in the college is _____.

• A. $1000$
• B. $975$
• C. $900$
• D. $225$

Comments

This question is sponsored by GATE 2010 TF: GA.8

Answer 1

Read question carefully. We cannot directly use the formula

$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B)  - P(B \cap C)  - P(C \cap A)  + P(A\cap B\cap C)$

because of the only keyword which means we do not directly know what is $P(A), P(B)$ or $P(C)$

Here, according to question statement, we can calculate total number of students belonging to at least one club : 
$P(A \cup B \cup C) = 60 + 80 + 30 + (40-2) + (12-2) + (7-2) + 2 =  225$

This is just $25\%$ of the total no of students since $75\%$ do not belong to any club.

So, total number of students$ = 225 \div 25\% = 225 \times 4 =  900.$

Comments

nice observation!

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Representation of Venn daigram is totally wrong???

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The answer given by @ijnuhb has the correct venn diagram

Answer 2

Let us solve it using the conventional formula :

(A U B U C) = A + B + C - (A ∩ B) - (B ∩ C) - (C ∩ A) + (A ∩ B ∩ C)

Notice that in this formula, 

A means students in Drama club (but not Drama club only).

B means students in Dance club (but not Dance club only).

C means students in Maths club (but not Maths club only).

So, A = students in Drama club ONLY + students in Drama and Dance clubs + students in Drama and Maths clubs - students in all clubs.

A= 60 + 40 + 7 - 2 = 105. 

Similarly,

B = 80 + 12 + 40 - 2 = 130.

C = 30 + 7 + 12 - 2 = 47.

Also, Given that (A ∩ B) = 40, (B ∩ C) = 12, (C ∩ A) = 7 &  (A ∩ B ∩ C) = 2

Now just substitute these values in the above formula.

(A U B U C) = 105 + 130 + 47 - 40 - 12 - 7 + 2 = 282 - 57 = 225.

But given that 75% of the students are not in any of these clubs ,i.e, 25% of the students are in at least one of these clubs.

0.25 * x = 225 ==> x = 900.

So, Total number of students in the college = 900.

Comments

this way is much easier to understand this question .Drawing venn diagram is easier one but here it was bit confusing.Only due to the word "only" made me to make a mistake

Answer 3

$900$, Hence option C is the answer.

Comments

the answer is 900. but in Pragy Sir's MyMarks app, the answer is given as 975. please correct it

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@Sukhbir Singh It is given as 900 . Check again 

Answer 4

Source: NPTEL

https://www.youtube.com/watch?v=7AjA2RqDou8

Watch from 38 minutes. 

Everything in GATE comes directly from NPTEL video lectures.

They have stopped making the NPTEL references for GATE but its alright.