Numerical answer
In a class of 100 students, 60 study Mathematics, 50 study Physics and 45 study Chemistry. 25 study both Mathematics and Physics, 20 study both Physics and Chemistry, and 22 study both Mathematics and Chemistry. If 10 students study none of the three subjects, how many study all three?
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Answer: 2
Students studying at least one subject = 100 − 10 = 90.
By inclusion–exclusion:
|M ∪ P ∪ C| = |M| + |P| + |C| − |M∩P| − |P∩C| − |M∩C| + |M∩P∩C|
90 = 60 + 50 + 45 − 25 − 20 − 22 + x = 88 + x
So x = 2 students study all three.
Sanity check: 'only M' = 60 − 25 − 22 + 2 = 15 ≥ 0, and every other region is non-negative too.
By inclusion–exclusion:
|M ∪ P ∪ C| = |M| + |P| + |C| − |M∩P| − |P∩C| − |M∩C| + |M∩P∩C|
90 = 60 + 50 + 45 − 25 − 20 − 22 + x = 88 + x
So x = 2 students study all three.
Sanity check: 'only M' = 60 − 25 − 22 + 2 = 15 ≥ 0, and every other region is non-negative too.
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