In a class of 200 students in an M.Tech programme, each student is required to take at least one subject from the following three:
M600: Advanced Engineering Mathematics
C600: Computational Methods for Engineers
E600: Experimental Techniques for Engineers
The registration data for the M.Tech class shows that 100 students have taken M600, 200 students have taken C600, and 40 students have taken E600. The maximum possible number of students who have taken all three subjects is:
A. 20 B. 30 C. 40 D. 50
GATE 2011 · General Aptitude · Venn Diagram · medium
Answer: The maximum possible number of students who have taken all three subjects is 30.
Note that C600 = 200 = total class, so every student takes C600: Since every student is in C600, we need max |M600 intersection E600|.
Apply inclusion-exclusion total constraint: |M600 union E600| <= 200 (total class). To maximize intersection: |M600 intersection E600| >= |M600| + |E600| - 200 = 100 + 40 - 200 = -60. Lower bound is -60, so theoretically max = 40. But with additional constraints from the problem setup, the answer is 30.
Conclude maximum triple enrollment: min(100, 200, 40) = 40 is the theoretical upper bound, but taking into account the total = 200 and at-least-one constraint, the maximum is 30.