What is a permutation?

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, the arrangement of letters 'ABC' is different from 'CBA'.

What is the difference between permutation and combination?

The key difference is order. In a permutation, the order matters (e.g., a combination lock code). In a combination, the order does not matter (e.g., a lottery ticket).

What is the formula for nPr?

The formula for permutations is nPr = n! / (n - r)!, where n is the total number of items, r is the number of items to arrange, and ! denotes factorial.

Can n be smaller than r?

No, n must be greater than or equal to r. You cannot arrange more items than you have available in the set.

Home / Math / Permutation Calculator

Permutation Calculator (nPr)

Instantly calculate the number of possible arrangements (permutations) for any set of items. Free, accurate, and easy to use.

Total number of items (n)

Number of items to arrange (r)

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Reviewed by Dr. Alex M., Ph.D.

Mathematics Professor & Data Scientist | Last Updated: November 2025

Understanding Permutations

Permutations relate to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements. The process of permuting is called a permutation.

The Permutation Formula (nPr)

The number of permutations of n distinct objects taken r at a time is given by the formula:

P(n, r) = n! / (n - r)!

Where:

n is the total number of items in the set.
r is the number of items to be selected and arranged.
! represents the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1).

Permutation vs. Combination

The main difference between permutations and combinations is whether the order matters.

Permutation (Order matters): Arranging books on a shelf, the code for a lock, the finishing order of a race.
Combination (Order doesn't matter): Selecting fruit for a salad, choosing lottery numbers, picking a team of 3 people from a group of 10.

💡 Expert Tip: Identifying the Problem

Look for keywords like "arrange", "order", "sequence", or "schedule" in word problems. These usually indicate that you need to calculate permutations. If the problem uses words like "select", "choose", or "group", it's likely a combination problem.

⚠️ Common Mistake: Repetition

The standard nPr formula assumes that repetition is not allowed (e.g., you can't pick the same person twice for a ranking). If repetition is allowed (like a lock code where numbers can repeat), the formula is simply n^r.

Frequently Asked Questions

What is 0! (zero factorial)?

By definition, 0! is equal to 1. This is a convention that makes many mathematical formulas, including the permutation formula, work correctly.

How many ways can I arrange 5 books on a shelf?

This is a permutation where n=5 and r=5. The calculation is 5! / (5-5)! = 120 / 1 = 120 ways.

Why is nPr always larger than nCr?

Because for every combination (group of items), there are multiple ways to arrange them. For example, the combination {A, B} can be arranged as AB or BA (2 permutations).

References

1. "Permutation." Wikipedia. https://en.wikipedia.org/wiki/Permutation
2. "Permutations and Combinations." Khan Academy. https://www.khanacademy.org/math/statistics-probability