Transparent method
The formula
Worked example
Example inputs
How to interpret the result
Whether order matters is the entire decision between these two formulas, and it's easy to answer wrong because the two situations can look superficially similar. Choosing 3 people from 10 for identical unnamed committee seats is a combination; choosing 3 people from 10 for president, vice president and secretary is a permutation, because swapping two people between those roles produces a genuinely different outcome.
Before you rely on it
What to check
Ask directly whether swapping two selected items changes the result. If it does, use permutations; if the result is identical either way, use combinations.
The common error
Where people go wrong with permutation & combination calculator
Defaulting to permutations because the formula sounds more thorough. Using it for a situation where order doesn't matter inflates the count, often substantially, because it separately counts arrangements a combination treats as the same outcome.
Sensitivity evidence
How n (total items) changes the permutations (order matters)
| n (total items) | Permutations (order matters) | Combinations (order doesn't matter) |
|---|---|---|
| 8 | 336 | 56 |
| 9 | 504 | 84 |
| 10worked example | 720 | 120 |
| 11 | 990 | 165 |
| 12 | 1,320 | 220 |
Every input, tested
Which input moves the permutations (order matters) most
| Input | Tested from | To | Permutations (order matters) at each end | Swing |
|---|---|---|---|---|
| r (items chosen) | 2 | 4 | 90 to 5,040 | 4,950 (688%) |
| n (total items) | 9 | 11 | 504 to 990 | 486 (68%) |
Two variables at once
Permutations (order matters) by n (total items) and r (items chosen)
| n (total items) \ r (items chosen) | 1 | 3 | 5 |
|---|---|---|---|
| 8 | 8 | 336 | 6,720 |
| 9 | 9 | 504 | 15,120 |
| 10 | 10 | 720 | 30,240 |
| 11 | 11 | 990 | 55,440 |
| 12 | 12 | 1,320 | 95,040 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| n (total items) | 10 |
| r (items chosen) | 3 |
| Permutations (order matters) | 720 |
| Combinations (order doesn't matter) | 120 |
Where the inputs come from
Enter n, the total number of items, and r, the number you choose, where r cannot exceed n. Use permutations when order matters, such as ranking finalists or arranging a password, and combinations when it does not, such as choosing a committee or a lottery ticket. The formulas use factorials: permutations are n factorial over n minus r factorial, and combinations divide that further by r factorial.
When the result stops applying
Repeated items, or choices with replacement, use different formulas. Very large values are shown approximately. Deciding whether order matters is the main source of mistakes, so restate the problem in words before choosing.
What to compare it with
Compute the chance of an outcome on the probability and binomial pages, and factorials on the factorial calculator. The expected value page evaluates bets, and the Bayes theorem calculator updates probabilities. The exponent page gives counts with replacement.
People also ask
Frequently asked questions
What is the difference between a permutation and a combination?
A permutation counts arrangements where order matters, and a combination counts selections where order does not.
How many ways can I choose 3 items from 10?
There are 120 combinations if order does not matter, and 720 permutations if it does, since each group of three can be arranged six ways.
When does order matter?
When different arrangements are different outcomes, such as first, second and third place, a PIN or seating positions.