Statistics Tool
Permutation Calculator
Calculate permutations P(n, r) with or without repetition. Get step-by-step factorial expansions and mathematical solutions.
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Interactive Permutation Calculator Workspace
Permutations Result210Formula Used: n! / (n - r)!
Factorial Expansion Steps
Step 1: Write the permutation formula: nPr = n! / (n - r)!
Step 2: Substitute values: 7P3 = 7! / (7 - 3)! = 7! / 4!
Step 3: Expand and simplify factorials: 7! / 4! = 7 × 6 × 5
Step 4: Multiply factors: 7 × 6 × 5 = 210
Mathematical Formula & Variables
Equation Model
nPr = n! / (n - r)!A permutation is an arrangement of items where the order of selection matters. nPr represents the number of unique ordered ways to select r items from a total set of n items.
Variable Definitions
| Symbol | Description |
|---|---|
| n | The total number of items in the set |
| r | The number of items to select |
| ! | Factorial operator (multiplication of all integers down to 1) |
How to Use the Permutation Calculator
- Enter the total number of items (n) in your set.
- Enter the number of items to choose (r) at a time.
- Enable "Allow Repetition" if items can be chosen multiple times (calculated as n^r).
- The calculator displays the step-by-step factorial expansions and the total count.
Practical Example Calculation
Scenario Context: Calculating how many ways a gold, silver, and bronze medal (r=3) can be awarded among 8 athletes (n=8).
Step 1: Write the formula: 8P3 = 8! / (8 - 3)! = 8! / 5!.
Step 2: Expand factorials: 8! = 8 * 7 * 6 * 5!.
Step 3: Simplify: (8 * 7 * 6 * 5!) / 5! = 8 * 7 * 6 = 336.
There are 336 unique ordered ways to award the medals.
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