The Inclusion–Exclusion Principle

Thirty students play cricket and twenty-five play football — so does "cricket or football" total $55$? Not if some play both. If ten play both, they have been counted twice, so the real total is $30 + 25 - 10 = 45$. That single correction is the whole principle: add everything, then subtract what you double-counted.

Why "Inclusion" and "Exclusion"

• Include every set by adding its size.
• Exclude the overlaps counted more than once by subtracting the intersections.

For three or more sets the signs keep alternating, because each correction slightly over-corrects and must itself be fixed.

The General Statement

For finite sets $A_1, A_2, \dots, A_n$,

Add all singles, subtract all pairwise intersections, add all triples, and so on — the sign alternates, starting with $+$.

Why It Works

An element lying in exactly $m$ of the sets is counted $\binom{m}{1}$ times among the singles, subtracted $\binom{m}{2}$ times among the pairs, added $\binom{m}{3}$ times among the triples, and so on. Its net count is

using $(1-1)^m = 0$ from the binomial theorem. So every element is counted exactly once.

A Three-Set Example

Of $100$ students: $40$ take Physics, $30$ Chemistry, $25$ Biology; $15$ take P and C, $10$ P and B, $8$ C and B; $5$ take all three.

So $67$ take at least one subject and $33$ take none. …