Exercise 479Combinatorics › Inclusion-exclusionHigh schoolAdded 3 Nov 2026

The three-set inclusion-exclusion formula

#inclusion-exclusion#three sets#union

Statement

In a survey of $100$ people, $60$ read $A$ , $50$ read $B$ , $40$ read $C$ , $30$ read $A$ and $B$ , $20$ read $B$ and $C$ , $15$ read $A$ and $C$ , and $5$ read all three. How many read at least one newspaper?

Hint

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

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Solution

Compute.
$60+50+40-30-20-15+5=150-65+5=90.$

Conclusion. $90$ people read at least one newspaper. $10$ read none.

The three-set inclusion-exclusion formula

Statement

Solution

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