Exercise · ZW-INCLUSION-EXCLUSION-PRINCIPLE-004 · ★★☆ Intermediate
How many integers from 1 to 100 are divisible by 2, 3 or 5?
Choose one option.
Show answer
- C. \(74\)
Explanation
With \(50,33,20\) and intersections \(16,10,6\) and \(3\): \(50+33+20-16-10-6+3=74\). Forgetting to add the triple term gives 71.
Review the concept: Inclusion-exclusion principle
It counts the elements of the union of finite sets \(A\), \(B\), \(C\), … by adding their sizes and correcting what was counted too often in the intersections.
\[\begin{aligned}|A\cup B|&=|A|+|B|-|A\cap B|\\|A\cup B\cup C|&=|A|+|B|+|C|\\&\quad-|A\cap B|-|A\cap C|-|B\cap C|\\&\quad+|A\cap B\cap C|\end{aligned}\]
More practice problems: inclusion-exclusion principle
- ZW-INCLUSION-EXCLUSION-PRINCIPLE-001 If |A|=12, |B|=9 and |A∩ B|=4, what is |A∪ B|?
- ZW-INCLUSION-EXCLUSION-PRINCIPLE-002 Of 30 students, 18 play soccer, 15 play basketball and 7 play both. How many play neither?
- ZW-INCLUSION-EXCLUSION-PRINCIPLE-003 How many integers from 1 to 100 are divisible by 2 or by 3?
- ZW-INCLUSION-EXCLUSION-PRINCIPLE-005 How many integers from 1 to 100 are divisible by neither 2 nor 3?
- ZW-INCLUSION-EXCLUSION-PRINCIPLE-006 Suppose |A|=10, |B|=8, |C|=6, |A∩ B|=3, |A∩ C|=2, |B∩ C|=1 and |A∩ B∩ C|=1. What is |A∪ B∪ C|?
- ZW-INCLUSION-EXCLUSION-PRINCIPLE-007 If |A∪ B|=|A|+|B|, what follows?
- ZW-INCLUSION-EXCLUSION-PRINCIPLE-008 How many binary strings of length 8 start with 1 or end with 00?
- ZW-INCLUSION-EXCLUSION-PRINCIPLE-009 If |A∪ B|=20, |A|=14 and |B|=11, what is |A∩ B|?
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