What is Combination & Permutation Calculator?

This calculator helps you solve counting problems by computing combinations (nCr) and permutations (nPr). Whether you're calculating lottery odds, determining possible arrangements, or solving probability questions, this tool provides instant results with step-by-step explanations.

Combination

C(n,r) - Order Doesn't Matter

Selecting r items from n items where arrangement is irrelevant.

Formula: C(n,r) = n! / (r! × (n-r)!)
Example: Choosing 3 people for a committee from 10 candidates
Result: ABC = BAC = CAB (same group)

Permutation

P(n,r) - Order Matters

Arranging r items from n items where sequence is important.

Formula: P(n,r) = n! / (n-r)!
Example: Assigning President, VP, and Secretary from 10 candidates
Result: ABC ≠ BAC ≠ CAB (different arrangements)

Key Relationship: P(n,r) = C(n,r) × r! — Permutations equal combinations multiplied by the number of ways to arrange the selected items.

How to Use

Choose Calculation Type

Click the Combination or Permutation tab based on whether order matters in your problem.

Enter n Value

Input the total number of items in the "Total items (n)" field.

Enter r Value

Input how many items you want to choose or arrange in the "Items to choose (r)" field.

View Result

The answer appears instantly as you type — no calculate button needed.

See Steps (Optional)

Click "Show calculation steps" to understand the mathematical process behind the result.

Input Requirements: Remember that r must be less than or equal to n, and both values must be non-negative integers.

Features

Instant Calculation

Results update automatically as you type, providing immediate feedback without clicking any buttons.

Step-by-Step Solutions

Expand calculation steps to see exactly how the answer was derived, perfect for learning.

Large Number Support

Calculate with n values up to 1000 using BigInt precision for accurate results.

Smart Formatting

Large results display in scientific notation for readability while maintaining precision.

Optimized Algorithms

Uses mathematical shortcuts like C(n,r) = C(n, n-r) for faster computation.

Quick Reference

Built-in comparison table of Combination vs Permutation with practical examples.

Input Validation

Clear error messages guide you when inputs are invalid, ensuring accurate calculations.

Frequently Asked Questions

What's the difference between combination and permutation?

In combinations, order doesn't matter (ABC = BAC = CAB). In permutations, order matters (ABC ≠ BAC ≠ CAB).

Use combinations for selecting groups (committees, lottery numbers, card hands)
Use permutations for arranging sequences (race positions, passwords, seating arrangements)

Why is my result showing in scientific notation?

When results exceed 20 digits, they're displayed in scientific notation (e.g., 1.234567e+50) for readability. The calculation remains precise using BigInt technology.

Example: C(100,50) produces a number with 29 digits, displayed as 1.008913e+29 for easier reading.

What does "n must be ≤ 1000" mean?

To ensure fast calculations, the maximum value for n is limited to 1000. This covers virtually all practical use cases while maintaining optimal performance.

Coverage: This limit handles everything from lottery calculations to complex statistical problems used in research and education.