Counting Principle
Multiply the number of choices at each step.
About this topic
The fundamental counting principle says that if one choice can be made in m ways and another in n ways, the pair can be made in m × n ways. An outfit from 4 shirts and 3 trousers gives 12 combinations. It extends to any number of independent steps.
A tree diagram lists the possibilities by branching at each stage, which is useful for small cases and for checking. For larger problems, multiply the number of options at each step. Restrictions are handled by choosing the restricted position first.
Example: a licence plate with 3 letters followed by 3 digits can be made in 26 × 26 × 26 × 10 × 10 × 10 = 17,576,000 ways. If letters cannot repeat, the count becomes 26 × 25 × 24 × 1000 = 15,600,000. A phone password of 4 digits has 10⁴ = 10,000 possibilities, which shows why longer passwords are much safer.
The principle underlies computer security, scheduling, genetics and the design of codes and puzzles. Tips: ask whether the stages happen one after another, whether repetition is allowed, and whether the order matters. Do not add the counts when both choices must be made; add only for either-or situations.
Key ideas
- Independent stages multiply
- Tree diagrams list outcomes
- 3 shirts × 2 pants = 6 outfits
Quick check
Use cases: where Counting Principle helps
- Homework: review the key idea behind Counting Principle before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Counting Principle without notes.
- Visual learning: play with the interactive lab to see how Counting Principle changes when you adjust its inputs.
- Teaching a friend: explain Counting Principle in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Counting Principle?
Multiply the number of choices at each step.
What will I learn in Counting Principle?
Independent stages multiply. Tree diagrams list outcomes. 3 shirts × 2 pants = 6 outfits.
Is Counting Principle part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Counting Principle lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Counting Principle?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.