Sets

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics and Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is .

Answer

10

Solution

Let x=n(M ); n(M )=40

40=20+25+16- [n(M )+n(P )+n(M ) ]+x

n(M )+n(P )+n(M )=21+x

x n(M ) 11

If x=11: n(M )=11 n(P )+n(M )=21

But each of these x=11 ⇒ sum 22, not possible ⇒ x 11

x=10 is possible (every condition holds):

[figure]

maximum =10

Answer: 10