Permutations vs Combinations: The Fundamental Counting Distinction
Both permutations and combinations count ways to select items from a group. The only difference: permutations care about order; combinations do not. This single distinction determines which formula to use.
Formulas
Permutations: P(n,r) = n! ÷ (n−r)!
Combinations: C(n,r) = n! ÷ [r!(n−r)!] also written ⁿCᵣ or C(n,r)Quick Decision Rule
- Use P: Passwords, race finishing positions, seating arrangements, PIN codes
- Use C: Lottery numbers, committee selection, card hands, sample groups
Worked Examples
- 5-digit PIN (digits 0–9, no repeat): P(10,5) = 10!÷5! = 30,240
- Lottery: pick 6 from 49: C(49,6) = 49!÷(6!×43!) = 13,983,816
- 4-player team from 10: C(10,4) = 210
- Gold/Silver/Bronze from 8 athletes: P(8,3) = 336
Relationship
P(n,r) = C(n,r) × r!
The r! factor converts combinations to permutations by counting all orderings of the selection.Calculate P and C: Free Permutation Calculator | Free Combination Calculator