A permutation counts ordered arrangements. Choosing a president, secretary and treasurer from a committee of ten gives 720 outcomes, not 120, because assigning the three roles differently produces a genuinely different result. The formula multiplies n by n−1 by n−2 and so on for r terms.
Permutations always outnumber combinations by exactly r factorial, since each unordered selection can be arranged in that many ways. A third case worth distinguishing is arrangement with repetition, where items can be reused — that gives n to the power r, which grows faster still and is the right model for passwords, PINs and combination locks.
The ordered and unordered counts compared table below carries 6 pre-computed rows, from n = 5, r = 2 to n = 52, r = 5, so a figure can be checked against this page without typing anything into it. It loads pre-filled, with total items (n) at 10, and the answer updates on every keystroke after that. The result is also read against 4 bands, from small at the low end to enormous at the high end, each with a note on what that range means in practice. Factorial Calculator and Combinations Calculator are the nearest neighbours to this one.
The formula
n- Total items available
r- How many are arranged
P(n, r)- Number of ordered arrangements
How it works, step by step
- Enter the total number of items available.
- Enter how many positions you are filling.
- The gauge shows the number of ordered arrangements.
- The unordered count and the with-repetition count appear below.
Worked examples
Three roles from ten people
P(10, 3) = 10 × 9 × 8 = 720. The equivalent combination count is 120, exactly 6 times smaller because the three roles can be swapped among any chosen trio.
Why repetition changes everything
A four-digit PIN allows repeats, giving 10 to the fourth = 10,000 codes. Without repetition there would be only P(10, 4) = 5,040, so allowing repeats adds 4,960 extra possibilities.
Checking the n = 20, r = 3 row
The reference table lists n and r = n = 20, r = 3, giving Permutations: 6,840, Combinations: 1,140, Ratio: 6×, With repetition: 8,000. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
What is the difference from a combination?
A permutation counts orderings; a combination does not. P(n, r) is always r factorial times larger than C(n, r).
What does P(n, n) equal?
Simply n factorial — the number of ways to arrange every item. Ten people can queue in 3,628,800 different orders.
Why is a combination lock badly named?
Because order matters on a combination lock, so it is really a permutation lock. The mathematical and everyday senses of the word have drifted apart.
How large can n be here?
Up to 170, beyond which the results exceed the range of double-precision arithmetic and would return infinity.
What do the bands under the permutations mean?
They label the figure so it can be read without a reference point of your own: Small from 0 to 100; Moderate from 100 to 10,000; Large from 10,000 to 1,000,000; Enormous from 1,000,000 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Permutations Calculator use?
It computes P(n, r) = n! ÷ (n − r)!, where n is total items available; r is how many are arranged; P(n, r) is number of ordered arrangements. The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
Permutations against combinations
| n and r | Permutations | Combinations | Ratio | With repetition |
|---|---|---|---|---|
| n = 5, r = 2 | 20 | 10 | 2× | 25 |
| n = 10, r = 3 | 720 | 120 | 6× | 1,000 |
| n = 10, r = 4 | 5,040 | 210 | 24× | 10,000 |
| n = 20, r = 3 | 6,840 | 1,140 | 6× | 8,000 |
| n = 26, r = 4 | 358,800 | 14,950 | 24× | 456,976 |
| n = 52, r = 5 | 311,875,200 | 2,598,960 | 120× | 380,204,032 |
The ratio between the two counts is always r factorial.