Probability
We extend to counting principles (fundamental counting principle, permutations, combinations) and apply these to compute probabilities of complex events.
8.1 Counting Principles, Permutations & Combinations
- Apply the fundamental counting principle
- Distinguish permutations (order matters) and combinations (order doesn't matter)
- Apply $nP_r=\frac{n!}{(n-r)!}$ and $nC_r=\binom{n}{r}=\frac{n!}{r!(n-r)!}$
Real-World Connection
A password with 6 different characters uses permutations — the order matters (AB is different from BA). Choosing a 5-person committee from 20 candidates uses combinations — only WHO is chosen matters, not the order they're listed.
Fundamental Counting Principle
If task 1 can be done in $n_1$ ways, task 2 in $n_2$ ways (independently), the combined task has $n_1\times n_2\times\ldots$ ways
Factorial
Number of ways to arrange $n$ distinct objects
Permutations
Ordered selections of $r$ from $n$ distinct objects
Combinations
Unordered selections of $r$ from $n$ distinct objects
💡 Tip
Ask yourself: 'Does order matter?' If rearranging would create a DIFFERENT outcome → permutation. If rearranging creates the SAME outcome → combination.
Worked Example
Counting with restrictions
Problem
Worked Example
Combination probability
Problem
CAPS Cognitive Level Distribution