Numerical answer
In a batch of 100 students, 60 study Mathematics, 50 study Physics and 40 study Chemistry. 25 study both Mathematics and Physics, 20 study both Physics and Chemistry, 15 study both Mathematics and Chemistry, and 10 study all three subjects. How many students study exactly one of the three subjects?
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Answer: 60
Only Maths = 60 − 25 − 15 + 10 = 30
Only Physics = 50 − 25 − 20 + 10 = 15
Only Chemistry = 40 − 20 − 15 + 10 = 15
Exactly one subject = 30 + 15 + 15 = 60.
Formula: Exactly one = ΣA − 2Σ(A∩B) + 3(A∩B∩C) = 150 − 2(60) + 3(10) = 60 ✓.
(Here the union is 150 − 60 + 10 = 100, so every student studies at least one subject.)
Only Physics = 50 − 25 − 20 + 10 = 15
Only Chemistry = 40 − 20 − 15 + 10 = 15
Exactly one subject = 30 + 15 + 15 = 60.
Formula: Exactly one = ΣA − 2Σ(A∩B) + 3(A∩B∩C) = 150 − 2(60) + 3(10) = 60 ✓.
(Here the union is 150 − 60 + 10 = 100, so every student studies at least one subject.)
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