Apply Pigeonhole Principle

Question

I found the following exercise online:

A group of students passed exams. $41$ passed Exam $A$, $28$ passed Exam $B$, $26$ passed Exam $C$. Also, $11$ passed both, $A$ and $B$, $10$ passed $A$ and $C$ and $12$ passed $B$ and $C$. So in total $69$ passed at least one test. How many passed all $3$?

I read recently a bit about the pigeonhole principle and wondered if you could apply it to this problem. If so, how?

Answer 1

I believe you do not need the pigeonhole principle here. Its a simple application of cardinality of intersection of sets.

You can find good information here.

But what you need in short is the following

$$ \vert a\cup b\cup c\cup\vert = \vert a \vert + \vert b \vert + \vert c \vert - \vert a \cap b\vert - \vert a \cap c\vert - \vert c \cap b\vert + \vert a \cup b \cup c\vert $$

Hope this helped