A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics & 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 be the number of students passing all three subjects. Testing x=11 (the cap from Mathematics-Physics) forces one of the remaining pairwise-only regions to be negative — impossible.
Testing x=10: a consistent split is Mathematics-only =9, Physics-only =15, Chemistry-only =5, (Mathematics ∩ Physics)-only =1, and the other pairwise-only regions 0 — this satisfies all three pairwise caps and the six regions plus the triple overlap sum to 9+15+5+1+0+0+10=40.
So the maximum number of students passing all three subjects is 10.
JEE Main · 2024 · 30 Jan 2024 · Shift 1 · Integer answer
