Math101 · Independent Ontario learning library
Math101Counting Principle
A precise guide to the fundamental counting principle, decision trees, restrictions, and probability denominators.
Precise definition
The fundamental counting principle says: if a process has stages with $n_1,n_2,\ldots,n_k$ choices and every choice at one stage leaves the stated number of choices at the next, then there are $n_1n_2\cdots n_k$ outcomes. Disjoint alternative cases are added rather than multiplied.
Notation and mathematical language
An outcome must be defined precisely enough to decide whether order and repetition matter. A tree diagram represents branches; the number of leaves is the count. If later options depend on earlier choices, use branch-specific products and add the terminal counts.
Conceptual picture
Multiplication counts ordered sequences of compatible choices. Each valid outcome must correspond to exactly one route through the stages. This one-to-one requirement prevents both omissions and double counting.
Fully worked example
Interpretation and application
Counting principles build sample spaces for codes, schedules, genetics, and surveys. A uniform probability model must be justified; counting alone does not make real-world categories equally likely.
