Permutation Calculator

Calculate permutations nPr, the number of ordered arrangements of r items chosen from n, with or without repetition. Shows the factorial steps, the matching combinations count and the exact result for large numbers. Example: 10P3 = 720.

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Permutations & Combinations Calculator

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Total items (n)

Items chosen (r)

Repetition

How to use the Permutation Calculator

Enter the total number of items n and how many you are choosing, r.
Choose whether repetition is allowed.
Press Calculate to get the exact count and the working.

Formula and method

Without repetition: nPr = n! / (n – r)!. With repetition: n^r. Compare combinations: nCr = n! / (r! (n – r)!), so nPr = nCr x r!.

Worked examples

Question	Calculation	Answer
Podium: 3 winners from 10 runners	10P3 = 10 x 9 x 8	720
Ways to arrange 5 books	5P5 = 5!	120
4-digit PIN, digits may repeat	10^4	10,000
Letters of ABC in order	3P3	6

Permutation versus combination

	Permutation	Combination
Order matters?	Yes	No
Formula	n! / (n-r)!	n! / (r!(n-r)!)
10 choose 3	720	120

Quick reference table nPr

n \ r	1	2	3	4
4	4	12	24	24
5	5	20	60	120
6	6	30	120	360
8	8	56	336	1,680

Frequently asked questions

What is a permutation?

An ordered arrangement of items chosen from a set.

What is the formula for nPr?

n! divided by (n – r)!.

What is 0!?

1, by definition.

When is repetition allowed?

When the same item can be used more than once, such as digits in a PIN; the count is n^r.

What is the difference between nPr and nCr?

nPr counts ordered selections; nCr ignores order, so nPr = nCr x r!.

Why is my number so large?

Factorials grow very fast; 20! already exceeds 2 quintillion.

Sources

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Effortless Math Team

10 minutes ago