A permutation is an ordered arrangement of objects. In a linear permutation, objects are arranged in a straight line and the order matters.
Special cases:
If each of positions can be filled by any of objects (repetition allowed):
If objects contain of one kind, of another, and of a third:
Example: Arrangements of letters in STATESMAN (9 letters, S×2, T×2, A×2):
When distinct objects are arranged in a circle, rotations of the same arrangement are considered identical. Fix one object's position and arrange the remaining :
Example: 5 people seated at a round table: ways.
When a condition limits one position (e.g., a number must be even), fill that position first, then fill the remaining positions.
Example: 3-digit even numbers from without repetition:
Treat the group of items that must stay together as a single entity. Then:
Example: 8 books on a shelf, 3 specific books always together:
To ensure certain items are never next to each other:
Example: 4 men and 3 women in a line, no two women adjacent:
Proof:
Permutations appear in many real-world contexts:
| Application | How Permutations Apply |
|---|---|
| Cryptography | Number of possible ordered keys/passwords from a character set |
| DNA sequences | Ordered arrangements of nucleotide bases (A, T, G, C) |
| Lottery odds | Counting ordered draws to estimate winning probability |
| Playlists | Number of ways to order a set of songs for an occasion |
Example (Password): A 4-character password using digits 0–9 with no repetition: