Math

nPr Calculator

Calculate Permutations Instantly

Enter n and r to compute P(n,r) = n!/(n-r)! -- the number of ways to arrange r items from n distinct items when order matters. The calculator shows step-by-step formula evaluation and compares the result with C(n,r) to illustrate the P = C * r! identity.

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Lottery Draw

Calculate the number of possible 6/49 lottery combinations.

Key values: Total balls: 49 · Drawn: 6 · Combination mode

Race Podium

Find the number of ways to award Gold, Silver, and Bronze among 8 runners.

Key values: Runners: 8 · Medals: 3 · Permutation mode

Secret Santa

Calculate valid gift assignments where nobody draws their own name.

Key values: Participants: 8 · Derangement mode

Password Keyspace

Estimate the number of possible 8-character passwords from printable ASCII.

Key values: Character set: 94 · Length: 8 · Repetition allowed

Combinatorics Calculator

Enter n and r to compute permutations, combinations, factorials, derangements, or Catalan numbers.

Documentation

This calculator is also known as nPr Calculator.

Read the complete guide

What Is nPr?

nPr (also written as P(n,r)) counts the number of ordered arrangements of r items from n distinct items. Unlike combinations, the order of selection matters: AB and BA are counted as different permutations.

Key Relationship

Permutations and combinations are related by a simple formula:

P(n,r) = C(n,r) * r! -- each combination has r! orderings
P(n,r) is always greater than or equal to C(n,r)
When r = 1, P(n,1) = C(n,1) = n

Examples

Race Podium: P(8, 3)

How many ways can 8 runners finish in Gold, Silver, and Bronze?

P(8, 3) = 8 * 7 * 6 = 336 possible podium arrangements.

Key takeaway: Order matters: Gold-Silver-Bronze is different from Silver-Gold-Bronze.

Working with Permutations

Tips for solving permutation problems:

Ask: does rearranging the selection give a different outcome? If yes, use permutations
Use the P vs C comparison panel to see how order multiplies the count
For permutations with repetition (e.g., passwords), use the Perm + Rep mode instead

Frequently Asked Questions about nPr Calculator

When should I use nPr instead of nCr?

Use nPr when the order of selection matters -- for example, awarding 1st, 2nd, and 3rd place medals, or arranging books on a shelf.

What is P(n, n)?

P(n,n) = n! -- the total number of ways to arrange all n items in a row. For example, P(4,4) = 4! = 24 represents all orderings of four distinct objects.

What does P(n, 0) equal?

P(n,0) = 1 for any n ≥ 0. There is exactly one way to arrange zero items: the empty arrangement. This follows directly from the formula: n! / (n-0)! = n! / n! = 1.