Permutations vs Combinations
Both count ways to select r items from a set of n items. The difference is whether order matters.
Permutation Formula (nPr)
Order matters. The number of ordered arrangements of r items from n:
nPr = n! / (n − r)!
Example: How many ways can 3 runners finish 1st, 2nd, 3rd from 8 contestants?
- 8P3 = 8! / 5! = 8 × 7 × 6 = 336
Combination Formula (nCr)
Order does not matter. The number of ways to choose r items from n:
nCr = n! / (r! × (n − r)!)
Example: How many ways to pick a 3-person committee from 8 people?
- 8C3 = 8! / (3! × 5!) = 336 / 6 = 56
Relationship Between nPr and nCr
Since combinations ignore order, nCr = nPr / r!. Every combination corresponds to r! permutations (all the rearrangements of the same r items).
Common Examples
| Scenario | Type | Formula | Result |
|---|---|---|---|
| Lottery: pick 6 from 49 | Combination | 49C6 | 13,983,816 |
| 4-digit PIN (0-9) | Permutation | 10P4 | 5,040 |
| Poker hand (5 from 52) | Combination | 52C5 | 2,598,960 |
| Podium (top 3 of 20) | Permutation | 20P3 | 6,840 |
| Team of 4 from 15 | Combination | 15C4 | 1,365 |
Special Cases
- nP0 = 1: There is exactly one way to arrange zero items
- nPn = n!: Arranging all items is the full factorial
- nC0 = nCn = 1: One way to choose nothing or everything
- nC1 = n: Choosing 1 item from n gives n options
- nCr = nC(n−r): Choosing r items to include is the same as choosing n−r to exclude
With Repetition
The formulas above assume items are selected without replacement. With repetition:
- Permutations with repetition: nr (e.g., 4-digit PIN with repeats: 10&sup4; = 10,000)
- Combinations with repetition: (n + r − 1)! / (r! × (n − 1)!)