The short answer
Permutations count arrangements where order matters (P(n,r) = n! ÷ (n−r)!); combinations count selections where order doesn't (C(n,r) = n! ÷ (r!(n−r)!)). Choosing a 3-place race podium from 10 runners is a permutation — P(10,3) = 720 different orderings. Choosing a 3-person committee from the same 10 people is a combination — C(10,3) = 120, since committee membership doesn't care about order.
Key takeaways
- Combinations are always smaller than (or equal to) the matching permutation, since C(n,r) = P(n,r) ÷ r! — dividing out all the ways to reorder the same group.
- C(n,r) = C(n, n−r) — choosing which r items to include is equivalent to choosing which n−r items to leave out.
- When r = n, both reduce to counting all n items: P(n,n) = n! and C(n,n) = 1.
- Factorials grow extremely fast, so P(n,r) and C(n,r) can become astronomically large even for modest values of n and r.
The permutation and combination formulas
| Concept | Formula | Order matters? |
|---|---|---|
| Permutation | P(n,r) = n! / (n−r)! | Yes |
| Combination | C(n,r) = n! / (r!(n−r)!) | No |
Worked example: race order vs. committee selection
With 10 runners (n = 10), choosing 3 (r = 3):
P(10,3) = 10 × 9 × 8 = 720 (gold/silver/bronze orderings)
C(10,3) = 720 ÷ 3! = 720 ÷ 6 = 120 (3-person committees)
Every group of 3 runners corresponds to 3! = 6 different podium orderings (who got gold, silver, bronze), which is exactly why the permutation count (720) is 6 times the combination count (120).
The symmetry rule for combinations
C(8,2) = C(8,6) = 28
Choosing 2 people out of 8 to include in a group produces the exact same count as choosing the 6 people to exclude — they're two ways of describing the same split. This symmetry, C(n,r) = C(n, n−r), can save calculation effort: if r is more than half of n, it's usually faster to compute C(n, n−r) instead.
Common mistakes to avoid
- Using the permutation formula when order genuinely doesn't matter — this inflates the true count by a factor of r!.
- Forgetting that r cannot exceed n — there's no way to arrange or select more items than exist in the group.
- Confusing the standard (no-repetition) formulas with "with repetition" versions, which apply to different problems, like a 4-digit PIN where digits can repeat.
- Losing track of r in multi-stage selection problems — forgetting to reduce the remaining pool after each stage already chosen.
Related calculators
- Probability Calculator — turn a permutation or combination count into an actual probability.
- Big Number Calculator — handle the huge factorials that large n and r values can produce.
- Scientific Notation Calculator — express very large permutation or combination results compactly.
- Statistics Calculator — apply counting results to broader statistical analysis.