Exercise 996Combinatorics › Inclusion-exclusionL1Added 3 Nov 2026

Exactly one property

#IE#exactly one#properties

Statement

Given $|A|=15$ , $|B|=20$ , $|C|=10$ , $|A\cap B|=5$ , $|A\cap C|=4$ , $|B\cap C|=3$ , $|A\cap B\cap C|=2$ , how many have exactly one property?

Hint

Sum singles, subtract twice pairwise, add three times triple.

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Formula. Exactly one $=\sum|A_i|-2\sum|A_i\cap A_j|+3|A\cap B\cap C|$ .

Compute. $45-2(12)+3\cdot2=45-24+6=27$ .

Conclusion. $27$ items have exactly one property.