Softonic

Permutations Calculator

Step 1 / 3
Next

A permutation counts ordered selections without repetition. Choosing first, second and third place from 10 runners gives P(10, 3) = 720 because changing the order creates a different result. Enter n, the number of available items, and r, the number of ordered positions.

How to calculate permutations

  1. 1

    Enter n (total items)

    The size of the pool you are choosing from.

  2. 2

    Enter r (items chosen)

    How many slots you are filling, in order.

  3. 3

    Check the range

    Use whole numbers with 0 ≤ r ≤ n ≤ 170. Invalid ranges return 0.

  4. 4

    Read P(n, r)

    The exact integer result appears below the inputs.

Formula used by this calculator

P(n, r) = n! / (n - r)!

This applies when order matters and an item cannot be selected twice. If repetition is allowed, the count is n^r; this calculator does not calculate that case.

If r = n (you are arranging every item), the standard formula collapses to n! - the classic factorial.

Permutations vs combinations

Concept Order matters? Formula
Permutations Yes n! / (n - r)!
Combinations No n! / (r! * (n - r)!)

Rule of thumb: if swapping two picks changes the outcome (1st/2nd/3rd on a podium), it is a permutation. If it does not (who is on the committee), it is a combination.

Real-world permutations

  • Tournament rankings. Ordering all 8 finalists: 8! = 40,320.
  • Seat arrangements. Seating 5 guests in 5 chairs: 5! = 120.
  • Medal positions. Awarding three distinct medals among 10 finalists: P(10, 3) = 720.

Factorial growth is explosive

n! grows quickly: 10! = 3,628,800, while 20! = 2,432,902,008,176,640,000. This calculator uses exact integer arithmetic and accepts n up to 170; it does not switch to scientific notation.

Frequently Asked Questions

Permutations care about order (first, second, third place); combinations do not (just “who made the team”). Picking 3 from 10: 720 permutations but only 120 combinations.

No. It calculates ordered selections without replacement. For r ordered positions where each of n items may be reused, calculate n^r instead.

There is exactly one way to arrange zero items - the empty arrangement. This is also why 0! = 1. It keeps the formulas consistent.

Use whole numbers with 0 ≤ r ≤ n and n no greater than 170. The result is shown as an exact integer.

Related Tools

BMI Calculator

Calculate body mass index from height and weight. Shows WHO category, healthy-weight range and limitations of BMI.

Social Security Calculator

Estimate a US Social Security retirement benefit from your PIA and claiming age. Compare monthly and annual amounts at 62, FRA 67 and 70.

Age Calculator

Calculate exact age in years, months and days from a birth date, plus total days, hours and the next birthday countdown.

Fahrenheit to Celsius Converter

Convert Fahrenheit temperatures to Celsius with the exact formula, Kelvin output and common reference points for weather, cooking and health.

Dog Age Calculator

Convert your dog's age to human years using modern veterinary research. Breed-size aware: small, medium and large dogs age differently.

Calorie Calculator

Estimate daily calorie needs for your goal using Mifflin-St Jeor BMR and activity factors. Includes deficit and surplus targets.

Tool available in other languages