When order matters
A permutation is an arrangement in which order matters. ABC and CBA are different permutations.
All n items
The number of ways to arrange n different items is n! (factorial).
Example: Arrange 5 books: 5! = 120.
r items chosen from n
nPr = n! / (n − r)!
Example: 8 runners. How many ways to award gold, silver and bronze?
8P3 = 8!/5! = 8 × 7 × 6 = 336
Think of it as filling positions: 8 choices for gold, 7 for silver, 6 for bronze.
Identical items
If some items are identical items, divide by the factorial of each repeated group.
n! / (a! b! c! …)
Example: Arrangements of TOOTH: 5 letters, T × 2, O × 2.
5!/(2! × 2!) = 120/4 = 30
Restrictions
Restriction: certain items together. Treat them as one block.
Example: 6 people in a row; Ana and Ben must sit together.
- Block (AB) + 4 others = 5 units → 5! = 120
- Ana and Ben can swap → × 2
- Total: 240
Restriction: certain items apart. Use total − together.
- Total: 6! = 720
- Apart: 720 − 240 = 480
Restriction: a fixed position. Place it first.
Example: 5-letter arrangements of ABCDE with A first: 4! = 24.
Circular permutations
Around a round table, rotations are the same arrangement:
(n − 1)!
Example: 7 people around a table: 6! = 720.
Permutation or combination?
| Ask | If yes |
|---|---|
| Does the order matter? (ranks, positions, codes) | permutation |
| Order doesn’t matter? (teams, groups, hands of cards) | combination |
Worked examples
Example 1: How many 4-letter "words" from A–Z with no repeats? 26P4 = 26 × 25 × 24 × 23 = 358,800.
Example 2: Arrangements of BANANA? 6!/(3! × 2!) = 60.
Example 3: 10 students; choose a president, VP and treasurer. 10P3 = 720.
Common mistakes
- Using combinations when order matters.
- Forgetting to divide for identical items.
- Forgetting the internal arrangements of a block (× 2!).
Try it yourself
- 6P2
- Arrangements of the letters in LETTER.
- 5 people in a row with Sam always at the left end.
- 8 people around a circular table.
- 4 people in a row with two specific people not side by side.
Answers: 1) 30 2) 180 3) 24 4) 5,040 5) 12
Quick recap
- A permutation counts arrangements where order matters: nPr = n!/(n − r)!.
- Divide by factorials for identical items.
- Circular permutations: (n − 1)!; handle each restriction carefully.