Q.In a class of $40$ students, $30$ passed in Accountancy and $25$ passed in Economics; $20$ passed in both subjects. How many students passed in at least one subject, and how many failed in both?

Let $A$ = passed Accountancy, $E$ = passed Economics. "At least one" is $n(A \cup E)$ by inclusion–exclusion; "failed both" is the rest of the $40$. …

Let $A$ be the students who passed Accountancy and $E$ those who passed Economics, with $n(A) = 30$, $n(E) = 25$, $n(A \cap E) = 20$, out of $40$ students.

Passed at least one:

Failed both = total minus those who passed at least one:

$40 - 35 = 5.$ …

Forgetting to subtract the $20$ who passed both leads to $55$, which absurdly exceeds the class size — a built-in check that the overlap must be removed. 'Failed bo …