A permutation counts arrangements in which order matters. It can be used to count ways to assign different positions, arrange selected objects, or choose a sequence from a larger group. Unlike combinations, permutations treat two selections in a different order as different outcomes.

Permutation formula

nPr = n! ÷ (n − r)!

Here, n is the total number of distinct items, r is the number of positions being filled, and ! means factorial. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Worked example

Suppose three different awards are given to three people chosen from five finalists, with each award going to a different person. The order of recipients matters because first, second, and third place are different roles. The number of arrangements is 5P3 = 5! ÷ 2! = 120 ÷ 2 = 60.

How to calculate a permutation

  1. Count the available distinct items to find n.
  2. Determine how many ordered positions are required to find r.
  3. Check that repetition is not allowed under the standard formula.
  4. Substitute n and r into n! ÷ (n − r)!.
  5. Simplify the factorials before multiplying when possible.

Permutation or combination?

Use permutations when changing the order creates a different result, such as assigning ranked places or arranging letters. Use combinations when only the group matters, such as choosing a committee with no assigned roles. The combination formula is nCr = n! ÷ [r!(n − r)!].

Assumptions and limitations

The standard nPr formula assumes distinct items and no repeated selection. Problems that allow repeated items or include identical objects require a different counting method. Factorials also grow quickly, so large calculations may be easier to evaluate with a calculator.

Frequently asked questions

Can r be greater than n?

No. For selecting distinct items without repetition, r must be between zero and n.

Does order matter in a permutation?

Yes. Two arrangements with the same items in different positions count separately.

What does n! mean?

It is the product of all positive integers from n down to 1, with 0! defined as 1.