The counting principle is a powerful shortcut for finding how many outcomes are possible when a process has several stages. Instead of listing every outfit, password, route, or meal combination, you multiply the number of choices at each step. This idea matters because many probability and combinatorics problems begin by counting the size of a sample space.
It turns messy branching situations into clear arithmetic.
Key Facts
- Fundamental Counting Principle: if stage 1 has a choices and stage 2 has b choices, then total outcomes = a × b.
- For n stages with k1, k2, ..., kn choices, total outcomes = k1 × k2 × ... × kn.
- A tree diagram represents choices as branches, and each complete path from start to finish represents one outcome.
- If the number of choices changes after earlier choices, multiply the choices available at each stage along the process.
- With replacement means choices reset, so repeated choices may be allowed, such as 10 × 10 × 10 for a 3-digit code using digits 0 to 9.
- Without replacement means choices decrease, such as 10 × 9 × 8 for a 3-digit code with no repeated digits.
Vocabulary
- Fundamental Counting Principle
- A rule stating that the total number of outcomes in a multi-stage process is found by multiplying the number of choices at each stage.
- Outcome
- One complete result of a process, such as one finished outfit or one full password.
- Stage
- One step in a sequence of choices, such as choosing a shirt, then pants, then shoes.
- Tree Diagram
- A branching diagram that shows choices at each stage and all possible complete paths.
- Sample Space
- The set of all possible outcomes for an experiment or decision process.
Common Mistakes to Avoid
- Adding the choices instead of multiplying them is wrong when each outcome is made by combining one choice from each stage. For 3 shirts and 2 pants, the total is 3 × 2 = 6, not 3 + 2 = 5.
- Multiplying choices that are not independent of earlier restrictions is wrong because later stages may have fewer options. If digits cannot repeat, a 4-digit code from 0 to 9 has 10 × 9 × 8 × 7 outcomes, not 10 × 10 × 10 × 10.
- Counting incomplete paths in a tree diagram is wrong because an outcome must include one choice from every required stage. Only paths that reach the final stage should be counted.
- Ignoring special rules such as leading zeros is wrong because they change the number of choices at a stage. A 3-digit number cannot start with 0, so the first digit has 9 choices, not 10.
Practice Questions
- 1 A cafeteria offers 4 sandwiches, 3 drinks, and 2 desserts. How many different meals can be made by choosing one of each?
- 2 How many 5-character codes can be made using the digits 0 to 9 if digits may repeat? How many can be made if digits may not repeat?
- 3 A student draws a tree diagram for 2 shirt choices, 3 pant choices, and 2 shoe choices, but counts only the first two levels and gets 6 outfits. Explain what is missing and find the correct total.