Permutations count arrangements where order matters. Combinations count selections where order doesn't matter.
Formulas
- Permutation: nPr = n! / (n−r)!
- Combination: nCr = n! / [r!(n−r)!]
Example
Choosing 3 from 10 people:
- Permutation (order matters — President, VP, Treasurer): 10P3 = 720
- Combination (committee of 3 — order doesn't matter): 10C3 = 120
nPr is always ≥ nCr because permutations count more arrangements.
A Memory Trick
If the word "arrangement" or "order" appears in the problem (ranking, seating order, PIN codes), it's a permutation. If the word "selection" or "group" appears (committees, hands of cards, teams), it's a combination. Asking "does swapping two chosen items create a new outcome?" is the fastest test — if yes, permutation; if no, combination.
Combinations With Repetition
Standard combination formulas assume you can't pick the same item twice. When repetition is allowed (like choosing 3 scoops of ice cream from 5 flavors, where you could pick the same flavor multiple times), a different formula applies: C(n+r−1, r) — a common source of errors when the "no repeats" assumption is applied incorrectly.