Permutation vs combination separates two counting situations: permutations count ordered selections, while combinations count unordered selections.
Counting language and notation
A selection of \(k\) items taken from \(n\) distinct items can treat arrangements as different outcomes (order matters) or treat arrangements as the same outcome (order does not matter). Factorials provide the standard notation for these counts: \[ n! = n(n-1)(n-2)\cdots 2\cdot 1,\quad 0!=1. \]
Permutation vs combination: “Order matters” leads to permutations; “order does not matter” leads to combinations.
Repetition rule: “Without repetition” means an item cannot be chosen twice; “with repetition” means it can.
Visualization of the decision structure
Formulas without repetition
When \(k\) distinct items are selected from \(n\) distinct items without repetition, the two core counts are:
| Model | Order | Count (formula) | Typical interpretation |
|---|---|---|---|
| Permutations | Order matters | \(\displaystyle {}_nP_k=\frac{n!}{(n-k)!}\) | k labeled positions filled by distinct items |
| Combinations | Order does not matter | \(\displaystyle {}_nC_k=\binom{n}{k}=\frac{n!}{k!(n-k)!}\) | Unlabeled group (committee/set) of size k |
A direct relationship connects the two counts: \[ {}_nP_k = {}_nC_k \cdot k!. \] The factor \(k!\) accounts for the number of internal arrangements of the same k chosen items.
Formulas with repetition
When repetition is allowed, the counting model changes because an item can appear multiple times in a selection.
| Model | Order | Repetition | Count (formula) | Typical interpretation |
|---|---|---|---|---|
| Ordered sequences | Order matters | Allowed | \(\displaystyle n^k\) | k positions, each position has n choices |
| Combinations with repetition | Order does not matter | Allowed | \(\displaystyle \binom{n+k-1}{k}\) | k items chosen from n types (multiset) |
Concrete comparison examples
A group of \(n=10\) students is available. Three distinct offices exist: president, vice president, and secretary. The outcome depends on which student holds which office, so order matters and repetition is not allowed: \[ {}_{10}P_3=\frac{10!}{7!}=10\cdot 9\cdot 8=720. \]
The same \(n=10\) students form a committee of size \(k=3\). Committee membership is what matters, so order does not matter and repetition is not allowed: \[ {}_{10}C_3=\binom{10}{3}=\frac{10!}{3!\,7!}=\frac{10\cdot 9\cdot 8}{3\cdot 2\cdot 1}=120. \] The relationship \( {}_{10}P_3 = {}_{10}C_3 \cdot 3! \) is visible numerically: \(720=120\cdot 6\).
A 4-digit access code uses digits 0–9 and allows repeated digits. Each of the \(k=4\) positions has \(n=10\) choices, so the count is: \[ 10^4=10000. \]
A selection of \(k=4\) donuts is taken from \(n=10\) donut types, allowing repeats, and only the multiset of types matters (order does not matter). The count is: \[ \binom{10+4-1}{4}=\binom{13}{4}=715. \]
Structural checkpoints for choosing the correct model
- Order sensitivity: Distinct roles, ranks, positions, or time order correspond to ordered outcomes.
- Group membership: Unlabeled groups, teams, committees, and sets correspond to unordered outcomes.
- Replacement rule: “Without replacement” corresponds to no repetition; “with replacement” corresponds to repetition.
- Internal rearrangements: The same chosen items produce \(k!\) different arrangements when order matters.
Common pitfalls in permutation vs combination
Confusion often appears when “different arrangements” are implicitly treated as the same. A committee of three members has one outcome per group, while officer assignments to those same three members have multiple outcomes because the roles create order.
Another frequent error is mixing “with repetition” into a no-repetition setting. Seating arrangements, role assignments, and drawing without replacement typically exclude repetition; codes, independent trials, and selection with replacement typically allow repetition.