Combination Calculator

Category: Algebra and General
- June 04, 2025
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Calculate combinations (nCr) and permutations (nPr) for mathematical problems, probability analysis, and statistical calculations. This calculator handles both small and large numbers with detailed explanations.

Basic Calculation

Total number of items to choose from
Number of items to select
Can items be selected multiple times?

Multiple Calculations

Starting value for range calculation
Ending value for range calculation

Probability Analysis

Total possible outcomes (e.g., deck of cards = 52)
Number of desired outcomes (e.g., aces = 4)

Advanced Options

Understanding Combinations and Permutations

Combinations and permutations are fundamental concepts in combinatorics used to count arrangements and selections in mathematics, probability, and statistics.

Key Concepts
  • Combinations (nCr): Number of ways to choose r items from n items where order doesn't matter
  • Permutations (nPr): Number of ways to arrange r items from n items where order matters
  • Factorial (n!): Product of all positive integers from 1 to n
  • Without Repetition: Each item can only be selected once
  • With Repetition: Items can be selected multiple times
Formulas
  • Combinations without repetition: C(n,r) = n! / (r! × (n-r)!)
  • Combinations with repetition: C(n+r-1,r) = (n+r-1)! / (r! × (n-1)!)
  • Permutations without repetition: P(n,r) = n! / (n-r)!
  • Permutations with repetition: P(n,r) = n^r
  • Relationship: P(n,r) = C(n,r) × r!
Common Applications
  • Card Games: Poker hands, bridge deals, lottery combinations
  • Statistics: Sampling methods, experimental design
  • Computer Science: Algorithm analysis, cryptography
  • Biology: Genetic combinations, species selection
  • Business: Team selection, product combinations
  • Sports: Tournament brackets, team formations
Pascal's Triangle
  • Each number is the sum of the two numbers above it
  • Row n contains the values C(n,0), C(n,1), C(n,2), ..., C(n,n)
  • Useful for expanding binomial expressions (a+b)^n
  • Appears in probability distributions and number theory

Disclaimer: This calculator provides exact mathematical results for combinations and permutations. For very large numbers, results may be displayed in scientific notation. Always verify calculations for critical applications and consider computational limitations for extremely large values.

What Is the Combination & Permutation Calculator?

The Combination & Permutation Calculator is a user-friendly tool that helps you calculate how many ways you can select or arrange a set of items. Whether you're forming a team, analyzing probabilities, solving math homework, or just curious about possibilities, this calculator gives quick and accurate answers.

Key Formulas Used:
  • Combinations without repetition: C(n, r) = n! / (r! × (n−r)!)
  • Combinations with repetition: C(n+r−1, r) = (n+r−1)! / (r! × (n−1)!)
  • Permutations without repetition: P(n, r) = n! / (n−r)!
  • Permutations with repetition: P(n, r) = n^r
  • Probability (exact): Favorable outcomes ÷ Total possible outcomes

How to Use This Calculator

Follow these steps to calculate combinations, permutations, and related probability values:

  • Enter the total number of items (n).
  • Enter how many items you want to choose or arrange (r).
  • Select if repetition is allowed or not.
  • Choose whether to calculate combinations, permutations, or both.
  • Optional: Enable range calculations or view Pascal’s Triangle for smaller values of n (up to 15).
  • Optional: Fill in sample space and favorable outcomes to calculate probabilities.
  • Click Calculate to get the result with step-by-step explanations.

What Makes It Helpful?

This calculator is ideal for anyone needing to:

  • Plan groupings or arrangements: Use it in games, events, or classroom activities.
  • Understand probability: Supports basic probability scenarios using combination and permutation logic.
  • Check work: Get clear calculation steps to verify math problems or assignments.
  • See patterns: Visualize values using Pascal’s Triangle or explore multiple r values at once.
  • Apply formulas: Helpful for learning or teaching concepts related to factorials and counting principles.

Real-World Uses

Here’s where this calculator comes in handy:

  • Card Games: Find the number of possible hands or winning combinations.
  • Lottery and Raffles: Determine odds of winning with different ticket combinations.
  • Business: Calculate the number of ways to form teams or product bundles.
  • Science and Genetics: Use in analyzing gene pairings or test case setups.

FAQ: Frequently Asked Questions

  • What is the difference between combinations and permutations?
    Combinations count how many ways you can choose items (order doesn’t matter). Permutations count arrangements (order matters).
  • Can I use this calculator for large numbers?
    Yes, but for values of n over 170, results may be limited due to factorial size limits. The calculator uses scientific notation for large outputs.
  • What is Pascal’s Triangle?
    A triangular array showing combination values. Each row represents all combinations for a given n.
  • Does it show the math steps?
    Yes. You can enable detailed steps to see how each value was calculated.
  • How is this different from a Scientific Calculator?
    While a scientific calculator supports advanced calculations and functions, this tool is focused specifically on permutations, combinations, and probability scenarios.

Related Math Tools You Might Find Useful

Here are Other calculators that work well with this one:

  • Percent Error Calculator: Learn how to calculate percent error and interpret results with a step-by-step error percentage guide.
  • Scientific Calculator: Handle complex math tools and trigonometric solver functions for deeper analysis.
  • Fraction Calculator: Simplify fractions, divide fractions, and convert values easily when working with ratios or probabilities.
  • Matrix Calculator: Perform matrix operations or use it as a matrix equation solver in Linear Algebra scenarios.
  • Exponent Calculator: Quickly solve exponents or powers needed for permutation formulas.

Summary

This calculator is a time-saving tool that simplifies how you calculate, understand, and apply permutations, combinations, and probability. It's useful for students, teachers, researchers, or anyone needing to measure outcomes from selections or arrangements. With its visual aids, detailed steps, and probability options, it bridges the gap between formulas and real-world usage.