When to use each
- Permutation (nPr) — order matters: medals in a race, 4-letter passwords, seat arrangements.
- Combination (nCr) — order doesn't matter: lottery numbers, committee picks, card hands.
- Factorial (n!) — total arrangements of n distinct items (= nPn).
Worked examples
| Question | Answer |
|---|---|
| How many 3-medal podiums from 10 runners? | 10P3 = 720 |
| How many 3-person committees from 10 people? | 10C3 = 120 |
| How many 5-card poker hands from a 52-card deck? | 52C5 = 2,598,960 |
| How many ways to arrange 7 books on a shelf? | 7! = 5,040 |
FAQ
Are repetitions counted?
No. nPr and nCr both assume each item is selected at most once (sampling without replacement). For sampling with replacement, the formulas are n^r and (n+r−1)Cr respectively.
Is 0! = 1?
Yes — by convention. It comes from the recursion n! = n × (n−1)! and from the gamma function definition Γ(1) = 1.