What is the difference between permutation and combination?

Permutation counts ordered arrangements (ABC and BAC are different). Combination counts unordered subsets (ABC = BAC). For the same n and r, permutations ≥ combinations.

When should I tick "Allow repetition"?

When the same item can be picked more than once — for example, four-digit PINs that may repeat digits. The formula becomes n^r.

Why does the calculator limit n to 5000?

To keep the response instant. For most everyday combinatorics problems, n well under 5000 is plenty.

Why is the answer shown as a giant integer, not scientific notation?

Floats lose precision past about 15 digits. BigInt keeps the answer exact regardless of size.

What if r > n without repetition?

There are not enough items to arrange — the calculator surfaces an error rather than returning a misleading zero.

Permutation Calculator

Compute P(n,r) = n! / (n−r)! or n^r with repetition. Exact integer answer up to n = 5000.

Written by Golam Rabbani, Founder & Lead Engineer

This interactive tool requires JavaScript. Read the formula and worked example below; you can compute the result by hand.

How to use this permutation calculator

Enter n (total items) and r (items to arrange).
Tick "Allow repetition" if items can repeat in the arrangement (uses n^r).
Press Calculate to get the exact integer count P(n,r) using BigInt arithmetic.
Copy the result or Reset to clear inputs.

About this permutation calculator

A permutation is an arrangement of r items taken from n distinct items where order matters. Without repetition, P(n, r) = n! / (n − r)! = n × (n − 1) × … × (n − r + 1). With repetition allowed, every position has n independent choices, so the count is n^r. The calculator uses JavaScript BigInt under the hood so very large permutations remain exact integers — never approximate or scientific-notation floats.

Worked example. How many ways can you arrange 3 books on a shelf from 10 candidate books? n = 10, r = 3 (no repetition). P(10, 3) = 10! / 7! = 10 × 9 × 8 = 720. So you can produce 720 ordered shelves. If books could be repeated (say, identical copies are stocked), the count becomes 10³ = 1,000. Permutations differ from combinations because order matters — for the same n = 10, r = 3 there are only C(10, 3) = 120 combinations.

FAQ

What is the difference between permutation and combination?: Permutation counts ordered arrangements (ABC and BAC are different). Combination counts unordered subsets (ABC = BAC). For the same n and r, permutations ≥ combinations.
When should I tick "Allow repetition"?: When the same item can be picked more than once — for example, four-digit PINs that may repeat digits. The formula becomes n^r.
Why does the calculator limit n to 5000?: To keep the response instant. For most everyday combinatorics problems, n well under 5000 is plenty.
Why is the answer shown as a giant integer, not scientific notation?: Floats lose precision past about 15 digits. BigInt keeps the answer exact regardless of size.
What if r > n without repetition?: There are not enough items to arrange — the calculator surfaces an error rather than returning a misleading zero.