Numarly

Permutation Calculator (nPr)

Calculate permutations — arrangements of items

What is it?

A permutation counts the number of ways to arrange r items chosen from a set of n items, where the order of selection matters.

Formula

nPr = n! / (n − r)!

Formula Explanation

Dividing n! (all possible orderings of n items) by (n−r)! removes the orderings of the items you didn't select, leaving only the count of ways to arrange the r chosen items in sequence.

Example Calculation

5P2 = 5! / 3! = 120 / 6 = 20 ways to arrange 2 items chosen from 5.

How to Use

  1. Enter the total number of items (n).
  2. Enter the number of items being arranged (r).
  3. Click Calculate nPr to see the number of possible arrangements.

Benefits

  • Instantly computes exact arrangement counts without manual factorial division.
  • Shows the formula substitution step for learning purposes.
  • Validates inputs to prevent invalid r > n calculations.

Use Cases

  • Counting the number of ways to award 1st, 2nd, and 3rd place in a race.
  • Password or PIN arrangement counting problems.
  • Probability problems where the order of outcomes matters.

What Your Result Means

The result is the exact number of distinct ordered arrangements possible when selecting r items from a set of n, where swapping the order of two selected items counts as a different arrangement.

Tips

  • Use permutation when order matters (like rankings); use combination when it does not (like groupings).
  • When r = n, nPr simplifies to n! (arranging all items).
  • When r = 0, nPr always equals 1 (there's exactly one way to arrange nothing).

Common Mistakes

  • Using permutation when the problem actually describes an unordered selection (that's a combination).
  • Entering r greater than n, which describes an impossible selection.
  • Forgetting that swapping two items in a permutation creates a distinct arrangement.

FAQs

How is permutation different from combination?

Permutation counts arrangements where order matters (like race finish positions), while combination counts selections where order doesn't matter (like choosing a team).

What does nPr mean?

nPr reads as "n permute r" — the number of ways to arrange r items selected from a total of n distinct items.

What happens if r equals n?

When r = n, nPr equals n! — the number of ways to arrange all n items in a sequence.

What is nPr when r = 0?

nPr equals 1 when r = 0, since there is exactly one way to arrange zero items (doing nothing).

Where are permutations used in real life?

Permutations are used in scheduling, ranking systems, password/combination lock counting, and any scenario where the sequence of selection matters.

Results are exact for n up to 170, limited by JavaScript number precision.

Last updated: July 25, 2026