What does this calculator do?

This permutation calculator finds the number of ways to choose and arrange r items from n distinct items when order matters. In a permutation, AB and BA are counted as different arrangements.

The calculator handles standard permutations without repetition. Each selected item is used at most once, and repeated-use arrangements are not included in this calculation.

What is a permutation?

A permutation is the number of ordered arrangements that can be made from a set of distinct items. It is often written as P(n,r) or nPr.

For example, if you arrange 2 letters from A, B, and C, the possible arrangements are AB, AC, BA, BC, CA, and CB. There are 6 arrangements because AB and BA are different when order matters.

What do n and r mean?

n is the total number of distinct items available. r is the number of items selected and arranged.

• n: total number of items.
• r: number of items to arrange.
• r cannot be greater than n.
• This calculator accepts integer values only.

For example, if 3 roles will be assigned from 8 students and the roles are different, then n = 8 and r = 3. Because the order or role assignment matters, this is a permutation problem.

Permutation formula

The standard permutation formula without repetition is P(n,r) = n! / (n − r)!. It can also be calculated as n × (n − 1) × ... × (n − r + 1).

The exclamation mark means factorial. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Example calculation: P(8,3)

If you want to arrange 3 items selected from 8 distinct items, the calculation is P(8,3).

• Formula: P(n,r) = n! / (n − r)!
• Substitution: P(8,3) = 8! / (8 − 3)!
• Simplified multiplication: 8 × 7 × 6
• Result: 336

Interpretation: 3 items chosen from 8 distinct items can be arranged in 336 different ordered ways.

Permutation vs combination

The key difference is order. In a permutation, order matters. In a combination, order does not matter.

• Permutation: AB and BA are different.
• Combination: AB and BA represent the same selection.
• Use permutation when positions, roles, ranks, or sequences matter.
• Use combination when you only need to choose a group.

For the same n and r values, the permutation result is usually larger than the combination result because each selection can have multiple orders.

How to read the example permutations

The calculator may show example permutation chips using automatic labels such as A, B, C, and D. This preview helps you understand what the result means.

For n = 4 and r = 2, the preview may show AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, and DC. AB and BA both appear because they are different ordered arrangements.

Valid inputs and boundaries

This calculator accepts integer values only. n must be between 0 and 200, and r must be between 0 and n.

• P(0,0) = 1.
• P(n,0) = 1, because there is one way to arrange no selected items.
• P(n,n) = n!, because all items are arranged.
• Fractional values, negative values, and r > n do not produce a normal result.

Common mistakes

• Using a combination formula when order actually matters.
• Using a permutation formula when order does not matter.
• Entering r greater than n.
• Trying to use decimal values for n or r.
• Expecting permutations with repetition or circular permutations.
• Assuming the example preview always lists every arrangement.
• Treating the calculator as a password generator.

Limitations of this calculator

This calculator is for standard permutations without repetition. It does not calculate permutations with repetition, circular permutations, multiset permutations, probability scenarios, lottery tools, or password generation.

The result is the mathematical formula result for supported integer inputs. The example preview is intentionally limited to 300 items and is not designed to list unlimited arrangements.