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

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.