Math101 · Independent Ontario learning library
Math101Fundamental Counting Principle
The fundamental counting principle multiplies the number of choices across successive stages to count complete outcomes efficiently.
If a process has several stages, multiply the number of available choices at each stage—after accounting for any restrictions.
The multiplication principle
If stage one can occur in $m$ ways and, for each of those, stage two can occur in $n$ ways, then the two-stage process has
possible outcomes. With more stages, continue multiplying the number of choices available at each stage.
Why multiplication works
Imagine a tree diagram. Each of $m$ first branches produces $n$ second branches, giving $n$ outcomes repeated across $m$ groups. The total is repeated addition:
The principle compresses a large tree without listing every leaf.
Worked example: independent stages
The stages need not be probabilistically independent; what matters is knowing the available count at each branch.
Repetition rules
Clarify whether choices may repeat. A four-digit PIN with repetition allowed has $10^4$ possibilities. If all digits must differ, the count is
If the first digit cannot be zero, its stage has only $9$ choices, while later stages may have different counts.
Common mistakes
Adding stage counts instead of multiplying. A complete outcome needs one choice from every stage.
Using the same count after a no-repetition choice. Available options decrease.
Treating overlapping cases as disjoint. This double-counts shared outcomes.
Forgetting a leading-zero restriction. Codes and numbers may have different rules.
Assuming order never matters. Define what makes two outcomes distinct.
Quick self-check
- What are the stages of one complete outcome?
- How many choices are available at each stage after earlier decisions?
- Is repetition allowed?
- Are alternatives disjoint before their counts are added?
- Would complement counting be simpler?
- Does the answer distinguish ordered and unordered outcomes correctly?
