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Mathematics · pattern A, Numeric fields

Permutations, where order matters.

Enter n and r for the number of ordered arrangements, with the combination count alongside so the difference between them is visible.

Inputs

The formula used

ⁿPᵣ = n! ÷ (n − r)!

Permutations count arrangements where order is significant. Combinations count selections where it is not, and are always fewer.

Permutations

720

n = 10, r = 3. Permutations count arrangements where order matters; combinations count selections where it does not.

Permutations ⁿPᵣ
720
Combinations ⁿCᵣ
120
n!
3,628,800
r!
6
(n − r)!
5,040
With repetition allowed
1,000

Worked steps

  1. 1ⁿPᵣ = n! ÷ (n − r)! = 3,628,800 ÷ 5,040 = 720

    arrangements of r items chosen from n, order significant

  2. 2ⁿCᵣ = ⁿPᵣ ÷ r! = 720 ÷ 6 = 120

    divide out the r! orderings of each selection

Order matters means dividing by less.

Choosing three from ten gives 720 ordered arrangements and 120 unordered selections, because each selection can be arranged in 3! = 6 ways. That factor of r! is the whole difference between the two formulas, and deciding whether order matters is the actual work.

Questions about permutation

When does order matter?
Passwords, race positions, seating and rankings. Committees, lottery draws and card hands are unordered.
What if items can repeat?
The count becomes nʳ, which the panel also reports.
Why is 0! equal to 1?
By definition, and because ⁿPₙ = n!/0! has to equal n!.

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