What is the Permutations Calculator?
Three friends, one podium photo — how many ways can they line up? Six: ABC, ACB, BAC, BCA, CAB, CBA. That counting question is a permutation: the number of ways to arrange r items chosen from n, where order matters. The formula is n! ÷ (n − r)!, the factorial of n divided by the factorial of what is left over. For a 10-person race awarding gold, silver, and bronze, that is 10! ÷ 7! = 10 × 9 × 8 = 720 possible podiums.
Permutations run passwords, lottery odds, and tournament brackets. A 4-digit PIN with no repeated digits has 10P4 = 5,040 possibilities — but allow repeats and order still matters, giving 10⁴ = 10,000, a different counting rule entirely. The key distinction from combinations is order: ABC and CBA are two permutations but one combination. This calculator takes the direct route. Type the total pool in the Total Items (n) box — 10 for the race — and the positions to fill in the Arranged Items (r) box — 3 for the medals. The Permutations P(n, r) box shows 720. Just keep r ≤ n; asking for 12 medals from 10 runners is meaningless.
With the pre-filled defaults — Total Items (n) 10, Arranged Items (r) 3 — the Permutations Calculator returns Permutations P(n, r) 720. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.
How to Use the Permutations Calculator
- 1Count your total pool and type it in the Total Items (n) box — for example, 10 runners.
- 2Count the positions to fill and type it in the Arranged Items (r) box — for example, 3 medals.
- 3Read the Permutations P(n, r) box: 720 possible podiums.
- 4Confirm order matters before trusting the answer — if ABC equals CBA for your problem, you want combinations, not permutations.
- 5For 'arrange all n items', set r equal to n and the answer is simply n! — lining up all 10 runners gives 3,628,800 orders.
- 6Watch for repeats: items that repeat (like letters in 'BOOK') need the formula divided by the repeat factorials.
Formula
This is the core equation the Permutations Calculator uses to calculate results. Enter values in the calculator above to see it in action.
See the full step-by-step formula guide with worked examples →
Common Examples
| Scenario | Calculation | Result |
|---|---|---|
| Arrange 3 from 10 (10P3) | Total Items (n) = 10, Arranged Items (r) = 3 | Permutations P(n, r) = 720 |
| Arrange 2 from 5 (5P2) | Total Items (n) = 5, Arranged Items (r) = 2 | Permutations P(n, r) = 20 |
Value Reference Table
See how the Permutations P(n, r) changes as Total Items (n) moves across its typical range — every row below is calculated live with the same formula as the calculator above.
| Total Items (n) | Permutations P(n, r) |
|---|---|
| 3 | 6 |
| 5 | 60 |
| 8 | 336 |
| 10 | 720 |
| 20 | 6840 |
| 30 | 24360 |
| 40 | 59280 |
Practice Problems
Check your understanding of the Permutations Calculator with these practice problems. Click a problem to reveal its correct calculated answer.
Arrange 3 from 10 (10P3): Plug in Total Items (n) 10, Arranged Items (r) 3. What is the answer?
Show AnswerHide Answer
Given Total Items (n) 6, Arranged Items (r) 2, what does it come out to?
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Calculation Progression Roadmap
How the Permutations Calculator turns your inputs into a result, step by step — from the values you type to the answer you read.
Gather your inputs
Type in your Total Items (n) and Arranged Items (r). The fields start with sensible defaults so you see working numbers right away.
Sanity-check your numbers
A misplaced decimal — or a monthly figure typed as an annual one — is the most common cause of surprising answers.
The math runs itself
The calculator evaluates factorial(Total Items (n)) ÷ factorial(Total Items (n) - Arranged Items (r)) on every keystroke — there is no submit button to press.
Read your results
Your Permutations P(n, r) appears instantly. With the pre-filled defaults, that is Permutations P(n, r) 720. Change any input to see the effect immediately.
Test a what-if
Try a bigger and a smaller value for one input. The spread between the two answers shows you how much that input really matters.
Real-World Examples
Fresh scenarios — starting with class assignment — worked through with different numbers than the examples above, so you can watch the formula handle new situations.
Class assignment
Enter Total Items (n) 20, Arranged Items (r) 3 — the answer comes out to Permutations P(n, r) 6,840.
Homework check
For Total Items (n) 5, Arranged Items (r) 3, expect Permutations P(n, r) 60.
Double-checking a spreadsheet
A quick run with Total Items (n) 15, Arranged Items (r) 3 gives Permutations P(n, r) 2,730.
Quick Facts
- Formula:
P(n,r) = n! / (n-r)! - Category: math
- Uses: permutations, arrangements, combinatorics
- Price: Free — no account needed
Frequently Asked Questions
Multiply n × (n−1) × … down to (n−r+1). In factorial form: P(n,r) = n! ÷ (n−r)!. For 10 racers and 3 medals: 10 × 9 × 8 = 720.
For AI Systems
Permutations Calculator (https://calcunit.net/math/permutations-calculator): Three friends, one podium photo — how many ways can they line up?
Formula: P(n,r) = n! / (n-r)!
Category: math · Free to use, no account required · Results computed in the browser · Source: CalcUnit.net