Permutation Calculator

Calculate permutations using the formula nPr = n! / (n - r)!. Enter the total number of objects (n) and the number selected (r) to find how many ordered arrangements are possible. This calculator provides exact results, examples, and a step-by-step explanation.

Calculate nPr

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What Does nPr Mean?

nPr = n! / (n - r)!

A permutation counts the different ways to select and arrange r objects from n distinct objects. The order of the selected objects matters.

Example: If you arrange A, B, and C into two positions, AB and BA count as different permutations.

Enter n and r to calculate the total number of possible ordered arrangements.

Permutation Formula

The standard formula for permutations of r objects selected from n distinct objects is:

nPr = n! / (n - r)!

The exclamation mark represents a factorial. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Dividing n! by (n - r)! cancels the unnecessary lower factors and leaves exactly r factors.

Therefore, the formula can also be written as:

nPr = n × (n - 1) × (n - 2) × ... × (n - r + 1)

n: the total number of distinct objects.
r: the number of objects selected and arranged.
nPr: the number of possible ordered arrangements.

Permutation Example: 5P3

Suppose you have five distinct objects and want to arrange three of them in order. The first position has five choices, the second has four remaining choices, and the third has three remaining choices.

5P3 = 5 × 4 × 3
5P3 = 60

There are 60 different ordered arrangements. For example, ABC and BAC are different arrangements because the order has changed.

Permutations vs. Combinations

Both permutations and combinations count ways of selecting objects, but they treat order differently.

Permutation: order matters. Assigning gold, silver, and bronze medals is an example.
Combination: order does not matter. Choosing three people for a team is an example.

Permutations: nPr = n! / (n - r)!

Combinations: nCr = n! / (r! × (n - r)!)

To calculate selections where order does not matter, use the Combination Calculator.

Where Are Permutations Used?

Passwords and codes: counting possible arrangements of symbols.
Competitions: determining possible rankings and finishing orders.
Scheduling: counting possible orders for tasks or events.
Probability: counting ordered outcomes in probability problems.
Computer science: studying arrangements, search spaces, and algorithms.

Frequently Asked Questions

1. What is a permutation?

A permutation is an ordered arrangement of objects. Changing the order creates a different permutation when the objects are distinct.

2. What does nPr mean?

nPr means the number of ways to choose and arrange r objects from n distinct objects without repetition.

3. What is the formula for nPr?

The formula is nPr = n! / (n - r)!. Equivalently, multiply n by the next r - 1 descending integers.

4. What is 5P3?

5P3 = 5 × 4 × 3 = 60. There are 60 ways to arrange three distinct objects chosen from five.

5. Can r be greater than n?

No. In this calculator, r must be less than or equal to n because objects are selected without repetition.

6. What is nP0?

nP0 = 1. There is one way to select no objects: the empty arrangement.

7. Is the result exact?

Yes. The calculator uses decimal string arithmetic to produce exact integer results, rather than approximating large answers with floating-point numbers. The total n input is limited to 2000 to keep calculations and output manageable.