Permutation & Combination Calculator
Quickly calculate permutations (nPr) and combinations (nCr) to figure out how many possible arrangements or selections exist for a given set of items. Enter your total items (n) and selected items (r) to see the results and formulas instantly.
Inputs
Results & Formulas
Permutations (nPr)
Formula: n! / (n - r)!
5! / (5 - 3)! = 5! / 2!
nPr = 60
Combinations (nCr)
Formula: n! / [r! × (n - r)!]
5! / [3! × (5 - 3)!] = 5! / [3! × 2!]
nCr = 10
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Understanding Permutations and Combinations
Probability and combinatorics are foundational pillars of mathematics, computer science, and statistics. Whether you are dealing with lottery odds, organizing a sports tournament, or breaking cryptographic codes, you will inevitably run into Permutations and Combinations. Our online Permutation and Combination Calculator provides a fast, error-free way to compute nPr and nCr, displaying the exact mathematical formulas used to derive the answer.
Permutations (nPr) – Order Matters
A permutation is a mathematical technique that determines the number of possible arrangements in a set when the order of the arrangements matters.
For example, if you have the letters A, B, and C, and you want to arrange 2 of them, "AB" and "BA" are considered two completely different permutations because the order is different. Think of a combination lock (which ironically should be called a permutation lock!)—the sequence 1-2-3 is not the same as 3-2-1.
nPr = n! / (n - r)!Combinations (nCr) – Order Does Not Matter
A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. You are simply choosing items to form a group.
Using the same letters A, B, and C, if you select 2 of them to form a team, the team "AB" is exactly the same team as "BA". Therefore, combinations will always yield a smaller number than permutations for the same set of variables.
nCr = n! / (r! × (n - r)!)Real-World Examples
Example 1: A Race (Permutations)
Imagine a race with 8 sprinters. You need to find out how many different ways the Gold, Silver, and Bronze medals can be awarded. Since winning Gold is different from winning Bronze, order matters.
n (total sprinters) = 8
r (medals) = 3
Calculation: 8P3 = 8! / (8 - 3)! = 8! / 5! = 8 × 7 × 6 = 336 possible ways.
Example 2: A Card Game (Combinations)
Suppose you are dealt a 5-card hand from a standard 52-card deck. The order in which you pick up the cards from the table doesn't matter; you still hold the same hand.
n (total cards) = 52
r (cards in hand) = 5
Calculation: 52C5 = 52! / (5! × (52 - 5)!) = 2,598,960 possible hands.
Why Use Our Calculator?
Calculating factorials by hand is tedious and practically impossible for large numbers (for instance, 10! is 3,628,800). Our tool handles the heavy lifting instantly. It prevents integer overflow, applies the correct mathematical formulas, and presents the step-by-step logic clearly so students can use it for homework verification.
