Permutation Calculator (nPr) with Steps

Home › Statistics & Probability Calculators › Permutation Calculator (nPr) with Steps

Permutation Calculator (nPr)

A permutation is an ordered arrangement. This calculator counts how many ways r items can be arranged when they are chosen from n distinct items, using P(n, r) = n! ÷ (n − r)!. Switch on repetition when the same item can be used more than once, as with digits in a PIN, and the count becomes nʳ.

Order matters in a permutation: choosing a president, vice-president and treasurer from 10 people gives P(10, 3) = 10 × 9 × 8 = 720 outcomes, because Ann–Ben–Cal is a different result from Cal–Ben–Ann. If only the group matters, divide by r! or use the combination calculator: C(10, 3) = 120.

Results are computed with exact integer arithmetic, so even P(300, 120) is returned digit for digit, and very long answers are also shown in scientific notation. Typical questions: race finishing orders, seating arrangements, passwords and lock codes, and arranging letters of a word with no repeated letters.

Frequently asked questions

What is the formula for permutations?

P(n, r) = n! ÷ (n − r)!. Without the factorials, multiply the r largest numbers down from n: P(10, 3) = 10 × 9 × 8 = 720.

What is the difference between permutation and combination?

Permutations count ordered arrangements; combinations count unordered groups. P(n, r) = C(n, r) × r!.

How many 4-digit codes are there with repetition?

10⁴ = 10,000, because each of the 4 positions can be any of the 10 digits.

What is nPr when r = 0?

1. There is exactly one way to arrange nothing.

Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.