Permutation and Combination Calculator
Calculate permutations and combinations for different values of n and r.
Understanding Permutation and Combination Calculator
The Permutation and Combination Calculator determines how many different ways items can be selected or arranged from a larger set. These calculations are important in combinatorics, probability, statistics, discrete mathematics, and many practical situations involving selections, rankings, schedules, and arrangements.
The key difference between a permutation and a combination is whether the order of the selected items matters. If changing the order creates a different outcome, use a permutation. If the same selected items represent the same outcome regardless of order, use a combination.
How to work through the calculation
Permutation and Combination Calculator is intended to show the mathematical relationship behind the answer, not just return a number. Enter the quantities requested by the selected method—such as Total Items (n), Items Selected (r), Calculation Type—and keep their units or mathematical meaning consistent. Where this calculator offers different shapes, methods, or forms, choose the one that matches the problem before entering the values.
How to interpret the result
For a useful check, look at the units and the scale of the answer before accepting it. A formula can be evaluated correctly while an input is still inappropriate for the problem. Repeating the calculation with a simple test value or checking the result against the expected direction of change can help identify an input mistake.
Permutation and Combination Calculator
The Permutation and Combination Calculator determines how many different ways items can be selected or arranged from a larger set. These calculations are important in combinatorics, probability, statistics, discrete mathematics, and many practical situations involving selections, rankings, schedules, and arrangements.
The key difference between a permutation and a combination is whether the order of the selected items matters. If changing the order creates a different outcome, use a permutation. If the same selected items represent the same outcome regardless of order, use a combination.
What Is a Permutation?
A permutation counts the number of ways to select and arrange r items from n total items when the order matters. This means that two arrangements containing the same items can still be different if those items appear in different positions.
For example, suppose three people are selected from a group of ten to receive first, second, and third place. The arrangement Alice-Bob-Carol is different from Bob-Alice-Carol because the positions are different. This makes the problem a permutation.
nPr = n! / (n-r)!
What Is a Combination?
A combination counts the number of ways to select r items from n total items when the order does not matter. The selected items form a group, so rearranging the same items does not create a new combination.
For example, if three people are selected from ten to form a committee, choosing Alice, Bob, and Carol is the same group as choosing Carol, Alice, and Bob. Since their order does not change the outcome, this is a combination.
nCr = n! / (r!(n-r)!)
Permutation vs. Combination
The easiest way to decide which calculation to use is to ask whether changing the order creates a different result.
Combination: Order does not matter
Ranking competitors, assigning seats, creating ordered passwords, or arranging objects are common examples of permutation problems. Selecting a committee, choosing lottery numbers, or selecting a group of items are examples where combinations are generally appropriate.
Understanding n and r
In both formulas, n represents the total number of available items, while r represents the number of items selected from that total. The calculator requires n ≥ r ≥ 0.
For example, if there are 10 available items and 3 are selected, then n = 10 and r = 3.
Factorial Notation
Both permutation and combination formulas use factorials. The factorial of a non-negative integer is the product of that number and every positive integer below it.
By definition, 0! = 1. This convention is important because it allows the standard permutation and combination formulas to work correctly when no items are selected or when all available items are selected.
How to Use the Calculator
Enter the total number of available items in n, then enter the number of items to be selected in r. Choose either Permutation (nPr) or Combination (nCr) depending on whether the order matters.
After calculating, the result shows the number of possible outcomes, the selected values, the applicable formula, calculation details, and an explanation of what the result represents.
Common Applications
Permutations and combinations are used in many areas. Permutations can be useful for rankings, seating arrangements, ordered selections, scheduling, and arranging objects. Combinations are commonly used for forming committees, selecting teams, choosing groups, probability calculations, and other situations where order is irrelevant.
Disclaimer
This calculator is intended for informational purposes only. While we strive for accuracy, we cannot guarantee that every result is correct for every situation.