Permutation Calculator

Count ordered arrangements of r items taken from n: P(n, r), also written nPr. Exact results for big numbers, with or without repetition, and the full list of arrangements for small cases.

P(10, 3)
720
Digits
3

Step-by-step working

  1. FormulaP(n, r) = n! ÷ (n − r)!
  2. The (n − r)! = 7! cancels, leaving the top 3 factorsP(10, 3) = 10 × 9 × 8
  3. Result720

Formula P(n, r) = n! ÷ (n − r)! with repetition: nʳ

Ask the tutor about your result

An AI tutor reads your inputs and results above and explains them in plain English. The numbers come from the calculator; the explanation is AI-written, so check it against your notes.

When order matters (and when it doesn’t)

Pick 3 of these 4 people for president, vice-president and secretary: that’s a permutation (order matters). There are 4 × 3 × 2 = 24 ways. But if you only pick a 3-person committee, every group of 3 appears 3! = 6 times in that list. So combinations = 24 ÷ 6 = 4.

  • Ana · Ben · Cai
  • Ana · Ben · Dee
  • Ana · Cai · Ben
  • Ana · Cai · Dee
  • Ana · Dee · Ben
  • Ana · Dee · Cai
  • Ben · Ana · Cai
  • Ben · Ana · Dee
  • Ben · Cai · Ana
  • Ben · Cai · Dee
  • Ben · Dee · Ana
  • Ben · Dee · Cai
  • Cai · Ana · Ben
  • Cai · Ana · Dee
  • Cai · Ben · Ana
  • Cai · Ben · Dee
  • Cai · Dee · Ana
  • Cai · Dee · Ben
  • Dee · Ana · Ben
  • Dee · Ana · Cai
  • Dee · Ben · Ana
  • Dee · Ben · Cai
  • Dee · Cai · Ana
  • Dee · Cai · Ben

The six highlighted rows are the same committee in different orders. Dividing by r! removes those duplicates: C(n, r) = P(n, r) ÷ r!.

Worked example

In a race of 8 runners, how many ways can gold, silver and bronze be awarded? 8 choices for gold, then 7 for silver, then 6 for bronze: P(8, 3) = 8 × 7 × 6 = 336.

With repetition allowed, each position has all n options: a 4-digit PIN has 10⁴ = 10,000 possibilities.

Arranging everything

Arranging all n items is P(n, n) = n!. Five books on a shelf can be ordered 5! = 120 ways. If some items are identical, divide by the factorial of each group’s size: the letters of MISSISSIPPI can be arranged 11! ÷ (4! · 4! · 2!) = 34,650 ways.

Frequently asked questions

Why does order matter here?

In a permutation, ABC and CBA are different outcomes. If they would count as the same, use the combination calculator.

What is P(n, 0)?

1: there is one way to arrange nothing.