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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 _____.

1. $1000$
2. $975$
3. $900$
4. $225$

edited | 3.4k views

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$= 225 \div 25\% = 225 \times 4 = 900.$

by Boss (10.7k points)
edited by
0

will you please explain more about why not to use this formula and various scenarios to avoid this.

0

@Asim Siddiqui 4 refer my answer :)

$900$, Hence option C is the answer.
by Boss (23.9k points)
edited by
0
the answer is 900. but in Pragy Sir's MyMarks app, the answer is given as 975. please correct it
0

@Sukhbir Singh It is given as 900 . Check again

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.

by Active (1.9k points)
edited
60 + (40-2) + 80 + (7-2) + 30 + (12-2) + 2 = 225 present in the clubs.
75 % are not in any of the clubs i.e. 25% are in clubs.
Therefore ,  x/4 = 225 => x =900
Ans is C
by (93 points)