Combination Calculator
Instantly calculate combinations (nCr) and permutations (nPr) for any set of items. Free, accurate mathematical tool for students, statisticians, and probability problems.
Calculate nCr & nPr
Enter the total number of items and the number of items to choose to get an instant, precise evaluation of combinations and permutations.
Set Details
Enter your values to calculate combinatorial possibilities
Mathematical Evaluation
Calculated combinations, permutations, and factorials
Combinatorics Formulas
Core mathematical formulas used in this calculator, using the example of n = 5 and r = 2.
| Concept | Formula | Description | Example (n=5, r=2) |
|---|---|---|---|
| Combinations (nCr) | n! / (r! × (n – r)!) | Selections where order does not matter. | 10 |
| Permutations (nPr) | n! / (n – r)! | Arrangements where order does matter. | 20 |
| Factorial (n!) | n × (n-1) × … × 1 | Product of all positive integers up to n. | 120 |
Combinatorics FAQ
Everything you need to know about combinations, permutations, and probability mathematics.
In a combination, the order of the selected items does not matter (e.g., choosing a team of 3 people from 10). In a permutation, the order does matter (e.g., awarding 1st, 2nd, and 3rd place to 3 people out of 10). Therefore, the number of permutations is always greater than or equal to the number of combinations for the same set of items.
The formula for combinations is nCr = n! / (r! × (n – r)!), where ‘n’ is the total number of items, ‘r’ is the number of items being chosen, and ‘!’ denotes a factorial (the product of all positive integers up to that number).
Combinations are used in probability to determine the total number of possible outcomes when the order of events or selections is irrelevant. This is essential for calculating odds in lotteries, card games, and statistical sampling.
By mathematical definition, 0! (zero factorial) is equal to 1. This convention ensures that formulas for combinations and permutations work correctly even when r = 0 or r = n, yielding a logical result of 1 possible way to choose nothing or everything.