What is a permutation?

A permutation is an arrangement where order matters. nPr counts how many different ways you can arrange r items chosen from a set of n. Awarding gold, silver, and bronze among ten athletes is a permutation problem because changing the order of the same three athletes produces a different outcome.

When should I use a permutation instead of a combination?

Use a permutation whenever the order of the chosen items matters and produces a distinct outcome. Arranging books on a shelf, ranking competitors, assigning different roles to people, and ordering a sequence of tasks are all permutation problems. If the order does not matter and you just want to know which items are selected, use a combination.

Permutation Calculator (nPr)

A permutation counts the number of ways you can arrange r items chosen from a set of n when the order of selection matters. This is one of the two fundamental counting tools in combinatorics, alongside combinations. The key distinction is that in a permutation, placing the same items in a different order gives a genuinely different outcome: awarding gold, silver, and bronze to three of five athletes is not the same as awarding bronze, gold, and silver to the same three athletes, so the order counts. The formula for nPr is n factorial divided by the factorial of n minus r, which in practice is the descending product of r terms starting at n. For five items taken three at a time, that is 5 times 4 times 3, giving 60 permutations. You enter n, the total number of items in the pool, and r, the number of items you are choosing and arranging, with r no larger than n. The calculator returns the permutation count, the expanded product form of the formula, and the corresponding combination count nCr for comparison, which shows how much larger the permutation count is because it includes all the different orderings of each group. Permutations grow very rapidly with n and r, and the calculator handles large values accurately. Use it for probability problems, combinatorics assignments, competition seeding, scheduling, and any situation where you need to count distinct ordered sequences. Enter your values and see the result appear immediately.

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Total items (n)

Items to arrange (r)

permutations (nPr)

Formula expansion5 × 4 × 3

Equivalent nCr10

nPr / nCr6 (= r!)

How it works

The calculator multiplies n by each descending 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 from the pool of n. The combination count nCr is calculated using the same product divided by r factorial, removing the duplicate orderings of each group. The ratio nPr divided by nCr always equals r factorial, confirming the relationship between the two operations.

Worked example

With n = 5 and r = 3, the formula gives 5 times 4 times 3 = 60 permutations. This might represent the number of ways to award first, second, and third place among five competitors. The equivalent combination count nCr is 10, because each group of 3 chosen from 5 can be arranged in 3! = 6 ways, and 10 times 6 equals 60.

Related calculators

Combination Calculator: count unordered selections where order does not matter.
Quadratic Formula Calculator: solve quadratic equations.
System of Equations Solver: solve two simultaneous linear equations.
Percent Error Calculator: measure deviation from a true value.

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