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The multiplication principle: if task A has m ways and task B has n ways, then doing A then B has m·n ways. The addition principle counts disjoint cases by summing. These foundational rules unlock all combinatorial counting.
Solve counting problems by breaking them into ordered steps. Recognize when to use multiplication (sequential choices) vs. addition (alternatives). Practice problems with restrictions and overlaps.
Multiplication applies when tasks are sequential and independent. Addition requires disjoint cases. Using both in the wrong context leads to over- or under-counting.
The multiplication principle formalizes an intuition you already have from counting-principles: when you make a sequence of independent choices, the total number of outcomes is the product of the options at each step. Suppose you're creating a username: 4 choices for a first letter and 10 choices for a trailing digit gives 4 × 10 = 40 possible usernames. The key word is *and* — you pick a letter *and* a digit. Independence matters: the number of digit choices can't depend on which letter you picked, or the product wouldn't be the right calculation.
The addition principle handles a different situation: mutually exclusive alternatives. You arrive at an intersection and must go either left *or* right. If the left road splits into 3 paths and the right splits into 5, there are 3 + 5 = 8 total routes — not 15, because you can't take both forks at once. Addition applies when the cases are disjoint: no outcome belongs to more than one case.
Most counting problems are built from these two rules in combination. To count the number of valid passwords that are either all-vowels (5 choices per character) or all-consonants (21 choices per character) for a 3-character password: count all-vowel passwords by multiplication (5 × 5 × 5 = 125), count all-consonant passwords (21 × 21 × 21 = 9261), then add the two disjoint cases (125 + 9261 = 9386). The structure is always the same: break the problem into cases, apply multiplication within each case (for sequential steps), apply addition across cases (for alternatives).
The most common error is applying multiplication to non-independent steps or addition to overlapping cases. If cases overlap, you'll double-count — which is exactly the problem that leads to the inclusion-exclusion principle you'll encounter next. The discipline of asking "are these cases disjoint?" and "are these steps independent?" before applying a rule is the core skill these two principles develop.