Exercise 1010Combinatorics › Inclusion-exclusionL1Added 3 Nov 2026

IE with complements

#IE#complements#sets

Statement

In a set $E$ of size $100$ , sets $A_1,A_2,A_3$ have sizes $40,50,60$ and pairwise intersections $20,25,30$ , triple intersection $10$ . Compute $|A_1^c\cap A_2^c\cap A_3^c|$ .

Hint

$|A_1^c\cap A_2^c\cap A_3^c|=|E|-|A_1\cup A_2\cup A_3|$ .

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Solution

Union. $40+50+60-20-25-30+10=85$ .

Complement. $100-85=15$ .

Conclusion. $15$ elements in none of the three sets.

IE with complements

Statement

Solution

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