Question : Directions: Among 80 students, 24 passed in Chemistry, 35 passed in Physics, and 29 passed in Mathematics. 8 students did not appear for the examination. If 2 students have passed in all three subjects, 2 students have passed in Physics and Chemistry both but not in Mathematics, 2 students have passed in Chemistry and Mathematics both but not in Physics, and 8 students have passed in Physics and Mathematics both but not in Chemistry, then how many students have passed in Mathematics only?

Option 1: 19

Option 2: 18

Option 3: 20

Option 4: 17

Correct Answer: 17

Solution : Given:
Use three circles representing the Venn diagram, where P denotes passed students of physics, C denotes passed students of chemistry and M denotes passed students of mathematics.
In the below Venn diagram, the intersection of all three circle areas shows the students who passed in all three subjects. The intersected area of the two circles shows the students who passed in both subjects respectively.
The outer circle shows the students who did not appear in the examination.
Students who did not appear for the examination = 8
Students passed in all three subjects = 2
Students passed in Physics and Chemistry but not in mathematics = 2
Students passed in Physics and Mathematics but not in Chemistry = 8
Students passed in Chemistry and Mathematics but not in Physics = 2
As total students who passed in Chemistry = 24
So, student who passed only in Chemistry = (24) – (2 + 2 + 2)
= 18
The total number of students who passed in Physics = 35
So, students who passed only in Physics = (35) – (2 + 2 + 8)
= 23
Similarly, the total number of students who passed in Mathematics = 29
So, students who passed only in Mathematics = (29) – (2 + 2 + 8)
= 17

So, the number of students who passed only in mathematics is 17. Hence, the fourth option is correct.