Count ordered arrangements with nPr = n!/(n−r)!—podium orders, letter codes, and shuffles. Pair with the combination calculator when order does not matter.
10P3 = ?
10 × 9 × 8 = 720 (order matters)
nPr vs nCr?
nPr counts order; nCr ignores it — see comparison below
With repetition?
Use n^r mode when items can repeat (e.g. 10^3 PINs)
A permutation calculator computes nPr—the number of ways to arrange r items from n when order matters—using nPr = n! ÷ (n−r)!, with exact integer results for modest n in your browser.
Permutations answer “how many ordered arrangements are possible?” If you line up 3 winners from 10 contestants, 10P3 counts every distinct podium order. If order did not matter, you would use combinations (nCr) instead.
The formula nPr = n! / (n−r)! builds the product r descending factors from n: 10P3 = 10×9×8 = 720. Factorials grow explosively, so this tool uses big integers and caps n at 500 for exact answers.
nPr = n×(n−1)×…×(n−r+1). When order does not matter, switch to nCr on the combination calculator.
Concise answers for common searches — definitions, steps, and comparisons.
5P2 = 5!/(5−2)! = 5×4 = 20 ordered arrangements.
Yes. All math runs locally in your browser; your inputs are not uploaded.
Permutations (nPr) count order. Combinations (nCr) count unordered groups—use the combination calculator when AB equals BA.
n is the pool size; r is how many positions you fill in order. Both must be whole numbers with 0 ≤ r ≤ n.
The tool shows nPr, the formula n!/(n−r)!, and factorial breakdowns for transparency.
Tap 5P2, 10P3, or 26P3 for common homework and password-length examples.
Copy the result line or share a link with your n and r encoded in the URL.
Input
n = 10 contestants, r = 3 placesOutput
10P3 = 72010×9×8 = 720 distinct gold-silver-bronze orderings.
Input
n = 26 letters, r = 3Output
26P3 = 17,576First letter 26 choices, then 25, then 24.
Input
n = 8, r = 8Output
8P8 = 8! = 40,320Every ordering of eight distinct items.
Common real-world scenarios where this tool saves time.
Compute nPr for ordered samples in finite populations without manual factorial tables.
Count possible podium or heat orders when only the sequence matters.
Estimate ordered character choices without replacement (simplified model—real policies vary).
Step-by-step chains that connect related tools for common tasks.
Pick the count that matches whether order matters.
| Question | Formula | 5 items, choose 2 |
|---|---|---|
| Permutation (order matters) | nPr = n!/(n−r)! | 5P2 = 20 |
| Combination (order ignored) | nCr = n!/(r!(n−r)!) | 5C2 = 10 |
| With replacement (ordered) | n^r | 5^2 = 25 (different model) |
This page computes nPr only. Use Combination Calculator for nCr.
Memorize-friendly anchors for quizzes.
| Notation | Value | Meaning |
|---|---|---|
| 5P2 | 20 | Two-place rankings from five |
| 10P3 | 720 | Three ordered picks from ten |
| 26P3 | 17,576 | Three letters A–Z, no repeat |
| nPn = n! | n! | Full permutations of n items |
Podium finishes and permutations of letters require nPr. Use nCr only when order is irrelevant.
You cannot arrange more distinct positions than available items—ensure r ≤ n.
Advertisement
nPr counts ordered arrangements: nPr = n! / (n−r)!. Example: 5P2 = 5×4 = 20. Read as “n permute r.”
Multiply r descending factors starting at n: nPr = n×(n−1)×…×(n−r+1). Equivalently compute n! and (n−r)! then divide.
Permutation cares about order (AB ≠ BA). Combination does not—nCr = nPr / r!. Use the combination calculator for unordered selections.
10P3 = 10×9×8 = 720. There are 720 ways to arrange 3 items chosen in order from 10.
5P2 = 5×4 = 20. Five choices for the first slot, four for the second.
This tool uses exact big integers for n up to 500. Beyond that, factorials overflow practical display—use scientific notation approximations elsewhere.
When each of r positions can be any of n items (repetition allowed), the count is n^r. Example: 10^3 = 1,000 three-digit PINs. Use the “With repetition” tab in this calculator.
Standard nPr assumes each item is used at most once (without replacement). If slots can repeat, switch to n^r mode—answers are usually larger.
No. Calculations run entirely in your browser.
n and r stay in your browser; permutation results are not uploaded to EverydayTools servers.
Uses exact bigint factorials for n ≤ 500. For probability coursework, confirm notation (nPr vs n^r with repetition) matches your problem statement.
More free tools for the same workflow.
Calculate combinations nCr = n!/(r!(n−r)!) for unordered selections. No upload: runs locally in your browser. Free, instant nCr results.
Calculate factorial n! for 0 ≤ n ≤ 500 with exact big-integer results. No upload—runs locally in your browser. Free, instant n! for combinatorics.
Calculate P(A), complements, P(A and B), and P(A or B) for independent events. No upload—runs locally in your browser. Free, instant.
Advertisement
Reviewed by EverydayTools Editorial Team on 2026-05-21.