About this calculator
The permutation and combination calculator is used to calculate the number of permutations and combinations. It is a basic tool in probability statistics and combinatorial mathematics. Arrangement takes into account the order of elements, while combination does not take into account order. For example, if you select 2 people from 3 people to line up, there are 6 permutations (AB, BA, AC, CA, BC, CB), but only 3 combinations (AB, AC, BC). Our free online permutation calculator provides a simple, fast and accurate solution.
The number of permutations P(n,r) represents the number of options for permuting r elements from n different elements. The formula is P(n,r) = n!/(n-r)!. The combination number C(n,r) represents the number of options for taking r elements from n different elements (regardless of the order). The formula is C(n,r) = n!/(r!×(n-r)!). The number of permutations is always greater than or equal to the number of combinations because permutations take order into account.
Using the permutation calculator is easy and intuitive. Just enter the total number n and the selection number r, click the calculation button, and you can immediately get the number of permutations and combinations. This tool is particularly suitable for students to learn probability and statistics, prepare for math exams, and analyze lottery probabilities.
What it calculates
Permutation and combination calculator is based on the complete Chinese reference article for this calculator. It explains what the tool calculates, when to use it, and how the result relates to the underlying formula.
Formula
Use the formula shown by Permutation and combination calculator together with the values entered in the calculator. Keep units consistent and check any restrictions before interpreting the answer.
- Identify the formula used by the calculator.
- Substitute the input values carefully.
- Simplify or interpret the result with the correct units.
Inputs
Enter the required values for Permutation and combination calculator. Use numeric inputs where requested, keep variable names consistent, and review the selected unit or calculation mode before calculating.
- Required numeric values.
- Relevant units or variable names.
- Calculation mode or target value when available.
Example
A typical example uses simple values so you can compare the input, formula, and output. This helps verify that the calculator is being used correctly.
| Step | What to check | Purpose |
|---|---|---|
| 1 | Enter sample values | Confirm how Permutation and combination calculator reads inputs |
| 2 | Review the formula | Understand the calculation method |
| 3 | Compare the result | Use the answer correctly |
How to interpret the result
The result should be read together with the formula, input values, and any displayed calculation steps. If the calculator shows multiple values, compare each label before using the answer.
Common mistakes
Most mistakes come from missing units, entering values in the wrong field, or ignoring formula restrictions. Recheck the inputs if the result looks unexpected.
- Check units and signs.
- Do not leave required inputs blank.
- Confirm that the formula conditions are satisfied.
How to use
Using the permutation calculator is easy. First, enter the total number n in the first input box, indicating how many different elements there are in total. For example, to select people from 10 people, n=10.
Then, enter the selection number r in the second input box, indicating how many elements to select. For example, select 3 people, r=3. Note that r cannot be greater than n. Click the "Calculate" button.
The calculator will immediately display the number of permutations P(n,r) and the number of combinations C(n,r). For example, P(10,3) = 720, C(10,3) = 120. This means that there are 720 ways to select 3 people out of 10 to line up, and 120 ways if the order is not considered. Because the numbers can be large, the results are displayed in scientific notation. Click the "Reset" button to clear all inputs and start a new calculation.
Main features
This permutation and combination calculator has the following features: simultaneously calculates the number of permutations and combinations; supports large number calculations (n ≤ 170); displays calculation formulas; uses scientific notation to display large numbers; automatically detects invalid input (r>n, etc.); has a simple and intuitive interface, easy to use; fast response speed, calculation results are displayed instantly; completely free, no registration or download required; supports desktop and mobile device access; suitable for students and mathematics enthusiasts.
Use cases
The permutation calculator is very useful in many scenarios. When students learn probability and statistics, permutation and combination are basic knowledge. For example, calculate lottery probabilities, poker card combinations, password possibilities, etc. In mathematics competitions, permutation and combination problems often arise.
In real life, it can be used to analyze the probability of winning the lottery. For example, if you choose 6 red balls from 33 red balls with two-color balls, the number of combinations C(33,6) = 1107568, that is, the probability of winning is about one in a million. In cryptography, the number of possible combinations of a password can be calculated. For example, for a 4-digit password (0-9), the number of permutations is P(10,4) = 5040.
In project management, task allocation plans can be calculated. For example, if 5 tasks are assigned to 3 people, the number of permutations P(5,3) = 60 ways. In sports competitions, the number of games played can be counted. For example, if 10 teams play in pairs, the number of combinations C(10,2) = 45 games. Whether in study, work or life, the permutation calculator is a useful tool.