Discussion
In a college 200 students took admission in different subjects. 56 students took Biology, 62 took Maths, 84 took Psychology. 18 students took Biology and Maths, 20 took Biology and Psychology and 12 took Maths and Psychology. If 8 students took all three subjects how many students took none of these subjects?
Answer: D
40 students took none of the three subjects.
Use the inclusion-exclusion rule for three sets to count students taking at least one subject.
\[ \begin{aligned} n &= 56 + 62 + 84 \\ &\quad - 18 - 20 - 12 + 8 \\ &= 202 - 50 + 8 = 160 \end{aligned} \]
So 160 students took at least one subject, and \(200 - 160 = 40\) took none.
Forgetting to add back the 8 who took all three gives 152 and 48 with none, which is not offered. Simply adding the three subject totals (202) exceeds the 200 students, which shows why the overlaps must be subtracted.
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