Counting: Inclusion–Exclusion & Stars and Bars

$|A\cup B\cup C| = |A|+|B|+|C| - |A\cap B| - |A\cap C| - |B\cap C| + |A\cap B\cap C|$.

In plain terms. Add each group, subtract the double-counted overlaps, then add back what you subtracted too many times.

Example. Integers $1$–$100$ divisible by $2,3,$ or $5$: $50+33+20-16-10-6+3 = 74$.

Nonnegative solutions of $x_1+\cdots+x_k=n$ number $\dbinom{n+k-1}{k-1}$; positive solutions number $\dbinom{n-1}{k-1}$.

In plain terms. Line up $n$ identical items and $k-1$ dividers; where the dividers go decides who gets what. Counting the arrangements is one binomial.

Example. Give $10$ candies to $3$ kids: $\dbinom{12}{2} = 66$ ways; if each must get one, $\dbinom{9}{2} = 36$.