Question

From 50 students taking examinations in Mathematics, Physics and Chemistry, each of the student has passed in atleast one of the subjects, 37 passed in Mathematics, 24 in Physics and 43 in Chemistry. At most 29 passed in Mathematics and Chemistry, at most 19 passed in Mathematics and Physics and at most 20 in Physics and Chemistry. Find the largest possible number of students that could have passed in all three examinations.

Answer 1

Let M be the set of students passing in Mathematics, P be the set of students passing in Physics and C be the set of students passing in Chemistry.

Given, $n(M\cup P\cup C)=50,n\left(M\right)=37\Rightarrow n\left(P\right)=24,n\left(C\right)=43,n(M\cap C)\le 29,n(M\cap P)\le 19,n(P\cap C)\le 20$

Now $n(M\cup P\cup C)=n\left(M\right)+n\left(P\right)+n\left(C\right)-n(M\cap P)-n(P\cap C)+n(M\cap P\cap C)\le 50$

$\Rightarrow 37+24+43-19-29-20+n(M\cap P\cap C)\le 50$

$\Rightarrow n(M\cap P\cap C)\le 50-36\Rightarrow n(M\cap P\cap C)\le 14$

Thus, the largest possible number of students that could have passed in all the three examinations is 14.