Exercise 1007Combinatorics › Inclusion-exclusionL1Added 3 Nov 2026

Partial three-set data

#IE#three sets#intersection

Statement

Given $|A|=20$ , $|B|=25$ , $|C|=15$ , $|A\cap B|=10$ , $|A\cap C|=8$ , $|B\cap C|=5$ , $|A\cup B\cup C|=45$ , compute $|A\cap B\cap C|$ .

Hint

Solve for the triple intersection using the IE formula.

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Formula. $|A\cup B\cup C|=\sum|A_i|-\sum|A_i\cap A_j|+|A\cap B\cap C|$ .

Plug in. $45=60-23+x$ , so $x=8$ .

Conclusion. $|A\cap B\cap C|=8$ .