What is a permutation?

A permutation is an arrangement where order matters. nPr counts how many ways you can arrange r items chosen from a set of n. Picking a first, second and third place from eight runners is a permutation, because the order of the placings matters.

What is the nPr formula?

nPr equals n factorial divided by the factorial of n minus r. In practice it is the product of n, n minus 1, and so on, for r terms. For example, 8P3 is 8 times 7 times 6, which is 336.

What is the difference between a permutation and a combination?

A permutation counts arrangements where order matters, while a combination counts selections where order does not. nPr is always greater than or equal to nCr for the same n and r.

Permutation Calculator (nPr)

This permutation calculator works out nPr, the number of different ordered arrangements of r items chosen from a set of n. Permutations are the right tool whenever order matters, which is more often than people expect. Working out how many ways first, second and third place can be awarded among eight runners, how many four digit PINs are possible without repeating a digit, or how many ways a committee can fill the distinct roles of chair, secretary and treasurer are all permutation problems. The formula is n factorial divided by the factorial of n minus r, which in plain terms is simply the product of n, then n minus 1, and so on for r terms. Because order is counted as different, the number of permutations grows very quickly, and even modest values of n and r produce large totals, so the calculator handles the multiplication for you and warns when the answer grows beyond the range it can show exactly. The key contrast to remember is with combinations, where order does not matter: choosing three people for identical prizes is a combination, but ranking them first, second and third is a permutation. Enter your n and r values, with r no larger than n, to see the result update as you type.

Total items (n)

Chosen and arranged (r)

336

ordered arrangements (nPr)

How it works

The calculator multiplies n by each smaller whole number down to n minus r plus 1, which is the same as n factorial divided by the factorial of n minus r. This counts every distinct ordering of the r chosen items. It computes the product directly rather than full factorials, so it stays accurate for larger inputs.

Worked example

For first, second and third place among eight runners, n is 8 and r is 3. That is 8 times 7 times 6, which equals 336 different ways to award the three placings.

Related calculators

Combination Calculator: order does not matter.
Factorial Calculator: n factorial.
Probability Calculator: probabilities.
Standard Deviation Calculator: spread of data.

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