Enumerative Combinatorics · Inclusion-exclusion, derangements, and surjections
Among thirty labeled projects, eighteen use method A, fourteen use method B, and twelve…
Problem
Among thirty labeled projects, eighteen use method \(A\), fourteen use method \(B\), and twelve use method \(C\). Eight use both \(A\) and \(B\), seven use both \(A\) and \(C\), and five use both \(B\) and \(C\); each of these pair counts includes projects that also use the third method. Three use all three methods. Count projects using at least one method, exactly one method, exactly two methods, and none. Derive the formulas by tracking how often a project with one, two, or three memberships is counted. Verify that the seven nonempty membership regions have nonnegative sizes consistent with all the data.
Hint
Track the contribution of one project according to how many methods it uses.
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