Exercise 243Combinatorics › Inclusion-exclusionHigh schoolAdded 3 Nov 2026

The two-set inclusion-exclusion formula

#inclusion-exclusion#two sets#union

Statement

In a class of $30$ students, $18$ study Spanish, $15$ study German, and $8$ study both. How many study at least one of the two languages?

Hint

$|A\cup B|=|A|+|B|-|A\cap B|$ .

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Formula. $|S\cup G|=18+15-8=25$ .

Conclusion. $25$ students study at least one language.

Corollary. $30-25=5$ study neither.