Permutations without repetition count orderings in which each item can be used only once. Awarding gold, silver and bronze to 8 finalists, choosing a president, secretary and treasurer from a club, or arranging 3 of 8 books on a shelf are all examples: once an item is placed it cannot be placed again, and swapping two items gives a different outcome.
The counting is a product of shrinking choices. The first position has n options, the second n − 1, and so on, for r positions. For 8 finalists and 3 medals that is 8 × 7 × 6 = 336. Arranging all n items gives n! – already 40,320 for n = 8.
Enter n and r, or type the items, and the calculator returns the exact count, compares it with the number of unordered groups and lists up to 300 arrangements. The chance of guessing the exact order correctly – useful for trifecta bets or ranking puzzles – is shown as well.
How to use the Permutation Without Repetition Calculator
- Number of objects to choose from (n) — Ignored when you type the objects themselves below.
- Positions to fill (r), each item used once
- Objects to list (optional, comma-separated) — If you type names here, n is taken from how many you entered and the list below uses them.
- List the selections
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The calculator multiplies the r factors n, n − 1, …, n − r + 1 in exact arithmetic. The listing generates arrangements in dictionary order, never reusing an item.
Worked example
A club of 8 elects a president, a secretary and a treasurer, with no one holding two posts. Choices: 8 for president, 7 for secretary, 6 for treasurer – 336 slates. Ignoring who holds which post there are 336 ÷ 3! = 56 groups of three officers.
Factorials
| n | n! |
|---|---|
| 3 | 6 |
| 5 | 120 |
| 8 | 40,320 |
| 10 | 3,628,800 |
| 13 | 6,227,020,800 |
Formula
P(n, r) = n! ÷ (n − r)! = n × (n − 1) × … × (n − r + 1)
All n items: P(n, n) = n!
Worked example
Inputs — Number of objects to choose from (n): 8; Positions to fill (r), each item used once: 3; Objects to list (optional, comma-separated): ; List the selections: Yes – show up to 300.
Results — Permutations without repetition = 336; Chance of guessing the exact order = 1 in 336 (0.297619%); Unordered groups C(n, r) = 56; Orders of each group (r!) = 6; For comparison: with repetition (nʳ) = 512; All arrangements = (A, B, C) (A, B, D) (A, B, E) (A, B, F)
(A, B, G) (A, B, H) (A, C, B) (A, C, D)
(A, C, E) (A, C, F) (A, C, G) (A, C, H)
(A, D, B) (A, D, C) (A, D, E) (A, D, F)
(A, D, G) (A, D, H) (A, E, B) (A, E, C)
(A, E, D) (A, E, F) (A, E, G) (A, E, H)
(A, F, B) (A, F, C) (A, F, D) (A, F, E)
(A, F, G) (A, F, H) (A, G, B) (A, G, C)
(A, G, D) (A, G, E) (A, G, F) (A, G, H)
(A, H, B) (A, H, C) (A, H, D) (A, H, E)
(A, H, F) (A, H, G) (B, A, C) (B, A, D)
(B, A, E) (B, A, F) (B, A, G) (B, A, H)
(B, C, A) (B, C, D) (B, C, E) (B, C, F)
(B, C, G) (B, C, H) (B, D, A) (B, D, C)
(B, D, E) (B, D, F) (B, D, G) (B, D, H)
(B, E, A) (B, E, C) (B, E, D) (B, E, F)
(B, E, G) (B, E, H) (B, F, A) (B, F, C)
(B, F, D) (B, F, E) (B, F, G) (B, F, H)
(B, G, A) (B, G, C) (B, G, D) (B, G, E)
(B, G, F) (B, G, H) (B, H, A) (B, H, C)
(B, H, D) (B, H, E) (B, H, F) (B, H, G)
(C, A, B) (C, A, D) (C, A, E) (C, A, F)
(C, A, G) (C, A, H) (C, B, A) (C, B, D)
(C, B, E) (C, B, F) (C, B, G) (C, B, H)
(C, D, A) (C, D, B) (C, D, E) (C, D, F)
(C, D, G) (C, D, H) (C, E, A) (C, E, B)
(C, E, D) (C, E, F) (C, E, G) (C, E, H)
(C, F, A) (C, F, B) (C, F, D) (C, F, E)
(C, F, G) (C, F, H) (C, G, A) (C, G, B)
(C, G, D) (C, G, E) (C, G, F) (C, G, H)
(C, H, A) (C, H, B) (C, H, D) (C, H, E)
(C, H, F) (C, H, G) (D, A, B) (D, A, C)
(D, A, E) (D, A, F) (D, A, G) (D, A, H)
(D, B, A) (D, B, C) (D, B, E) (D, B, F)
(D, B, G) (D, B, H) (D, C, A) (D, C, B)
(D, C, E) (D, C, F) (D, C, G) (D, C, H)
(D, E, A) (D, E, B) (D, E, C) (D, E, F)
(D, E, G) (D, E, H) (D, F, A) (D, F, B)
(D, F, C) (D, F, E) (D, F, G) (D, F, H)
(D, G, A) (D, G, B) (D, G, C) (D, G, E)
(D, G, F) (D, G, H) (D, H, A) (D, H, B)
(D, H, C) (D, H, E) (D, H, F) (D, H, G)
(E, A, B) (E, A, C) (E, A, D) (E, A, F)
(E, A, G) (E, A, H) (E, B, A) (E, B, C)
(E, B, D) (E, B, F) (E, B, G) (E, B, H)
(E, C, A) (E, C, B) (E, C, D) (E, C, F)
(E, C, G) (E, C, H) (E, D, A) (E, D, B)
(E, D, C) (E, D, F) (E, D, G) (E, D, H)
(E, F, A) (E, F, B) (E, F, C) (E, F, D)
(E, F, G) (E, F, H) (E, G, A) (E, G, B)
(E, G, C) (E, G, D) (E, G, F) (E, G, H)
(E, H, A) (E, H, B) (E, H, C) (E, H, D)
(E, H, F) (E, H, G) (F, A, B) (F, A, C)
(F, A, D) (F, A, E) (F, A, G) (F, A, H)
(F, B, A) (F, B, C) (F, B, D) (F, B, E)
(F, B, G) (F, B, H) (F, C, A) (F, C, B)
(F, C, D) (F, C, E) (F, C, G) (F, C, H)
(F, D, A) (F, D, B) (F, D, C) (F, D, E)
(F, D, G) (F, D, H) (F, E, A) (F, E, B)
(F, E, C) (F, E, D) (F, E, G) (F, E, H)
(F, G, A) (F, G, B) (F, G, C) (F, G, D)
(F, G, E) (F, G, H) (F, H, A) (F, H, B)
(F, H, C) (F, H, D) (F, H, E) (F, H, G)
(G, A, B) (G, A, C) (G, A, D) (G, A, E)
(G, A, F) (G, A, H) (G, B, A) (G, B, C)
(G, B, D) (G, B, E) (G, B, F) (G, B, H)
(G, C, A) (G, C, B) (G, C, D) (G, C, E)
(G, C, F) (G, C, H) (G, D, A) (G, D, B)
(G, D, C) (G, D, E) (G, D, F) (G, D, H)
(G, E, A) (G, E, B) (G, E, C) (G, E, D)
(G, E, F) (G, E, H) (G, F, A) (G, F, B)
(G, F, C) (G, F, D) (G, F, E) (G, F, H)
(G, H, A) (G, H, B) (G, H, C) (G, H, D)
(G, H, E) (G, H, F) (H, A, B) (H, A, C)
(H, A, D) (H, A, E) (H, A, F) (H, A, G)
… showing the first 300 of 336.
Key terms
- Permutation without repetition
- An ordered arrangement of distinct items, each used once.
- Factorial
- n! = n × (n − 1) × … × 1.
- Falling factorial
- n(n − 1)…(n − r + 1), another name for P(n, r).
Frequently asked questions
How many ways can 8 runners take gold, silver and bronze?
P(8, 3) = 8 × 7 × 6 = 336.
What is the formula for permutations without repetition?
P(n, r) = n! ÷ (n − r)!.
How many ways can 5 people sit in a row?
5! = 120.
How is it different from permutations with repetition?
With repetition each position has all n choices (nʳ); without, the choices shrink by one each time.
What is the chance of picking the first three finishers in order?
1 ÷ P(n, 3). For 8 runners: 1 in 336, about 0.30%.
What if r = 0?
There is exactly one empty arrangement, so P(n, 0) = 1.