The Fundamental Counting Principle is a simple rule for finding how many total outcomes are possible when a process has several stages. It matters because many probability and statistics problems begin by counting the size of a sample space. Instead of listing every outcome one by one, you multiply the number of choices available at each stage.
This makes large counting problems easier to organize and solve.
Key Facts
- Fundamental Counting Principle: Total outcomes = choices at stage 1 × choices at stage 2 × choices at stage 3 × ...
- If a process has m choices followed by n choices, then the total number of ordered outcomes is m × n.
- For k independent stages with c1, c2, ..., ck choices, total outcomes = c1 × c2 × ... × ck.
- With replacement means the same option can be used again, so the number of choices often stays the same at each stage.
- Without replacement means used options are removed, so the number of choices usually decreases at later stages.
- Permutations count ordered arrangements, while combinations count selections where order does not matter.
Vocabulary
- Fundamental Counting Principle
- A rule that finds the total number of outcomes by multiplying the number of choices at each stage of a process.
- Outcome
- One possible result of a choice process or experiment.
- Sample Space
- The set of all possible outcomes for an experiment or situation.
- Permutation
- An arrangement of items where the order of the items matters.
- Combination
- A selection of items where the order of the items does not matter.
Common Mistakes to Avoid
- Adding choices instead of multiplying them. Add only when choosing between separate cases, but multiply when stages happen together in a sequence.
- Treating dependent stages as independent. If one choice changes the number of later choices, update the count at each stage instead of reusing the same number.
- Ignoring whether order matters. Use ordered counting for passwords, rankings, and arrangements, but use combinations when only the group selected matters.
- Double counting outcomes in overlapping cases. When splitting a problem into cases, make sure the same outcome cannot appear in more than one case unless you subtract the overlap.
Practice Questions
- 1 A lunch menu has 4 sandwiches, 3 drinks, and 2 desserts. How many different lunches can be made by choosing one sandwich, one drink, and one dessert?
- 2 A 4-digit passcode uses digits 0 through 9. How many passcodes are possible if digits may repeat? How many are possible if digits may not repeat?
- 3 A student says there are 5 + 4 + 3 = 12 ways to choose an outfit from 5 shirts, 4 pants, and 3 pairs of shoes. Explain the mistake and give the correct counting method.