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Free Permutation & Combination (nPr / nCr) Calculator

Combinatorics is a cornerstone of discrete mathematics, probability, computer science, and statistics. Our free permutation and combination calculator computes permutations (nPr) where order matters, and combinations (nCr) where order does not matter. Solve ncr calculator n choose k problems and npr calculator permutations instantly with full factorial breakdowns.

Permutation & Combination (nPr / nCr) Calculator

Compute permutations (nPr) and combinations (nCr) for n choose r

Total Items (n)

Sample Chosen (r)

Combinations & Permutations

nCr (Combinations): 120

nPr (Permutations): 720

Formula: 10C3 = 10! ÷ [3! × (10-3)!]

Mathematical Formulas & Calculation Methods

Permutation Formula (nPr)

Calculates total ordered arrangements of r items chosen from n items using the npr calculator permutations formula.

nPr = n! ÷ (n - r)!

Combination Formula (nCr)

Calculates total unordered subsets of r items chosen from n items using the ncr calculator n choose k formula.

nCr = n! ÷ [ r! × (n - r)! ]

Factorial Formula (n!)

Product of all positive integers less than or equal to n using the factorial combination solver.

n! = n × (n - 1) × ... × 1 (with 0! = 1)

Worked Step-by-Step Examples

Example 1: Calculating Combinations (10 Choose 3)

"A committee of 3 people is selected from a group of 10 candidates. How many combinations are possible? <strong>how to calculate ncr step by step</strong>."

Steps:

n = 10, r = 3.
nCr formula: 10! ÷ [ 3! × (10 - 3)! ] = 10! ÷ (3! × 7!).
Simplify: (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120.

Result: There are 120 possible committee combinations.

Example 2: Calculating Permutations (5 P 2)

"How many ways can 1st and 2nd place prizes be awarded to 5 contestants?"

Steps:

n = 5, r = 2.
nPr formula: 5! ÷ (5 - 2)! = 5! ÷ 3!.
Simplify: 5 × 4 = 20.

Result: There are 20 ordered permutation arrangements.

What is Calculadora de Permutation & Combination (nPr / nCr)?

A permutation and combination calculator evaluates how many distinct groups or arrangements can be created from a set of objects.

Use Permutations (nPr) when order is important (e.g. PIN codes, race finishes). Use Combinations (nCr) when order is irrelevant (e.g. lottery picks, team selections).

How to Use Calculadora de Permutation & Combination (nPr / nCr)

1Enter total item count (n) and sample selection count (r).
2Select Permutation (nPr) or Combination (nCr) mode.
3View instant total arrangements, subsets, and factorial breakdown.
4Copy final answers directly into homework or project files.

Key Features

Dual calculation modes for both nPr and nCr.
Step-by-step factorial simplification display.
Handles large values of n and r with high precision.
Intuitive input validation preventing r > n errors.
100% free client-side browser execution.

Benefits of Using Calculadora de Permutation & Combination (nPr / nCr)

Saves hours manually simplifying long factorial expressions.

Eliminates arithmetic mistakes on combinatorics exams.

Essential tool for computer science algorithm design and statistics.

Works on all mobile devices and desktop browsers.

Practical Use Cases

Math Students

Solve combinatorics homework and master calculate npr and ncr online.

Computer Scientists

Analyze algorithm time complexity and subset generation.

Event Organizers

Determine group seating arrangements and tournament matchings.

Lottery & Game Analysts

Calculate exact odds for draw games and card hands.

Frequently Asked Questions

What is the key difference between nPr and nCr?

nPr measures ordered arrangements where sequence matters. nCr measures unordered combinations using the ncr calculator n choose k formula.

What is 0 factorial (0!)?

By mathematical definition, 0! equals 1.

Why is nCr equal to nC(n-r)?

Choosing r items to keep is mathematically identical to choosing (n - r) items to leave behind.

Is this combinatorics calculator free?

Yes, this permutation and combination calculator is 100% free.

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