Probability
Fundamental Counting Principle – The Multiplication Rule
The tables and trees from the last two pages both led to the same pattern: multiply the number of choices at each stage together. That pattern has a name – the Fundamental Counting Principle: if one event can happen in m ways, and a second, independent event can happen in n ways, then both events together can happen in m × n ways. With three or more independent events, simply keep multiplying.
This principle is the single most important idea in this entire section – permutations, combinations, and every other formula you'll meet are really just the Fundamental Counting Principle applied to specific, common situations.
Applying the Fundamental Counting Principle
If independent events can happen in m, n, p, … ways, the number of ways they can all happen together is m × n × p × …
5 × 8 × 4 = 160.
Real-Life Application
- Licence plates: the total number of possible plates is found using this principle.
- Product customisation: the number of possible product configurations (colour, size, material) multiplies out this way.
- Genetics: the number of possible gene combinations from two parents follows this same multiplication rule.
Key Takeaways
- The Fundamental Counting Principle multiplies the number of ways for each independent event.
- It generalises to any number of events, not just two.
- Nearly every combinatorics formula is a special case of this principle.