set theory

Question

from 50 students taking exam in mathematics, physics and chemistry, each of the student has passed in at least one of the subjects, 37 passes mathematics, 24 physics and 43 chemistry. At most 19 passed mathematics and physics, at most 29 mathematics and chemistry and at most 20 physics and chemistry. What is the largest possible number that could have passes in all the three subjects?

Answer 1

The first statement in the problem says in the set of all students the subset student pass no exam is an empty set, or it has 0 elements in that set.

 

The total number of students is 50.

Let

A ={students that pass math},

B ={students that pass physics},

C ={students that pass chemistry}.

 

Then

A∩B={students that pass math and physics},

B∩C={students that pass physics and chemistry},

C∩A={students that pass chemistry and math}.

and,

A∩B∩C={students that pass math, physics and chemistry}.

 

The notation |A| means the number of students that pass math. Then, |A| = 37.

From the first statement

|A∪B∪C|=50, that is the total number of students passing tests, and each student passed at least one set.

 

There is a formula for the number of elements in a set with three sets.

|A∪B∪C| = |A| + |B| +|C| - |A∩B| - |B∩C| - |C∩A| + |A∩B∩C|

     50     =  37 + 24  + 43 -   19    -20      -  29      + |A∩B∩C|

We solve that equation for |A∩B∩C|:

     50 = 36+|A∩B∩C|

     |A∩B∩C| =50 -36,

     |A∩B∩C| =14.

 

At most 14 students passed math, chemistry and physics.