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Principle of Inclusion and Exclusion

Count to 100. How many of those numbers are odd or multiples of 5? Since 50 are odd and 20 are multiples of 5, at first glance the answer is 70...

Principle of Inclusion and Exclusion Problem Solving

After a grueling nationwide examination, it was announced that
70% of students passed the Mathematics examination, and
60% of students passes the Science examination.

What is the minimum percentage of students that passed both examinations?

In a class of students, $$64\text{%}$$ of the students passed the physics exam and $$68\text{%}$$ of the students passed the math exam. What is the minimum percentage of students in the class that passed both exams?

How many numbers in the set $$\{1,2,\ldots,100\}$$ are divisible by $$5$$ or $$7?$$

A party of $$370$$ vile creatures is marching through the woods. Each of the creatures is either an orc, a goblin, or a half-orc/half-goblin. If there are $$206$$ creatures that are at least partly goblin, and $$211$$ creatures that are at least partly orcs, how many of the creatures are half-orc/half-goblins?

Consider all $$4$$-digit positive numbers using all of the digits $$\{1, 2, 3, 4\}$$. How many of these numbers have even first or second digit?

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