Award rankings
How many ways can gold, silver, and bronze be awarded to 3 people from a group of 10 finalists? This is a permutations problem because each position has a different meaning.
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Permutations Calculator
Calculate ordered arrangements with and without repetition using exact permutation formulas. Useful for rankings, code generation, slot assignments, probability setup, and combinatorics practice.
Permutations Calculator
Use this free permutations calculator to count ordered arrangements from a larger set. It supports permutations without repetition and permutations with repetition, making it useful for ranking problems, code generation, slot assignments, sampling, probability setup, and combinatorics practice. If you want to compare order-sensitive results against unordered counts, continue with the Combinations Calculator, or explore more tools in Math & Statistics Calculators.
Count ordered arrangements instantly
Enter n items and r positions to calculate how many different sequences are possible when order matters.
Useful for
Ranking and sequencing
Great for podiums, passwords, seat orders, codes, and placement problems.
Inputs
Use whole numbers only. In standard mode, n is the total number of items and r is the number of ordered positions you fill.
Example: awarding gold, silver, and bronze from 10 finalists uses n = 10 and r = 3.
Permutation count
720
Standard permutations count ordered arrangements without reuse. Different positions produce different outcomes.
Formula used
nPr = n! / (n - r)!
Exact result
720
Calculation steps
- 1Start with n = 10 and r = 3
- 2Use the formula nPr = n! / (n - r)!
- 3Multiply descending terms: 10 × 10 - 1 × ... for 3 positions
- 4The exact permutations result is 720
- 5If order did not matter, the combinations count would be 120
Interpretation
The same selection counted as combinations would be 120 because order would no longer matter.
What this permutations calculator does
This tool counts ordered arrangements, which makes it useful whenever positions, ranking, sequence, or arrangement order changes the meaning of the result. That includes podium finishes, passwords, seat plans, product-slot orders, and many classroom counting problems.
How permutations work
Permutations multiply the number of choices available for each position. Without repetition, the number of remaining options decreases after each pick. With repetition, every position still has the full set of n options, which leads to the simpler n^r formula.
Permutations formulas explained
Without repetition
nPr = n! / (n - r)!
With repetition
n^r
n
Total distinct items available.
r
Number of ordered positions filled.
nPr
Ordered arrangements without repetition.
n^r
Ordered arrangements when repetition is allowed.
Permutations calculator examples
Codes and passkeys
If a 4-position code can use 5 symbols and repeats are allowed, permutations with repetition give the total number of possible codes instantly.
Order-sensitive sampling
Permutations are often used before probability calculations when the order of drawing, assigning, or placing items affects the event definition.
If you want to compare this order-sensitive count with unordered selection counts, continue with the Combinations Calculator. If you want to turn arrangement counts into event likelihoods next, the Probability Calculator is the natural follow-up.
Why permutations matter
Permutations prevent undercounting when position or order changes the outcome. They are a core counting tool in probability, scheduling, assignment problems, ranking, and combinatorics.
When repetition changes the result
Allowing reuse keeps the number of choices constant across positions, which can make the result much larger. That is why permutations with repetition use n^r instead of nPr.
Math & statistics workflow
Permutations often work alongside combinations and probability. Use them to define an arrangement space first, then follow with probability or summary-statistics tools when the task expands.
Related tools
Frequently asked questions
It counts how many ordered arrangements are possible when choosing r items from n options. Because order matters in permutations, ABC and CBA are treated as different outcomes.
Explore This Tool in Context
Permutations Calculator is part of the Math & Statistics Calculators collection. If you want a broader view of similar workflows, open the Math & Statistics Calculators category page or browse all QuickTools categories.
Common next steps after this tool include Random Number Generator, Mean Calculator and Median Calculator.
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