A class has 40 students. 22 study Mathematics, 18 study Science, and 15 study English. 8 study Math and Science, 5 study Math and English, and 4 study Science and English. 2 study all three. How many

Here's the solution for :

Let $M$ be the set of students who study Mathematics, $S$ for Science, and $E$ for English.
Given:
Total students = 40
$|M| = 22$
$|S| = 18$
$|E| = 15$
$|M \cap S| = 8$
$|M \cap E| = 5$
$|S \cap E| = 4$
$|M \cap S \cap E| = 2$

1. How many study at least one subject? This is equivalent to finding $|M \cup S \cup E|$. We use the Principle of Inclusion-Exclusion.