Combination Calculator

Calculate combinations and selections

Combination Calculator
Select the type of calculation
Choose calculation approach
Total number of items
Number of items to select
Note: This calculator computes combinations, permutations, and factorial with step-by-step solutions.
Quick Examples:
About Combination Calculator & Visualizations
Combination Calculator

This advanced combination calculator provides comprehensive solutions for calculating combinations, permutations, and factorial with step-by-step explanations, making it essential for combinatorics, probability, and statistics.

Combination Types:
  • Standard (nCr): n! / (r!(n-r)!)
  • With Repetition: (n+r-1)! / (r!(n-1)!)
  • Multiset: n! / (r1!r2!...rk!)
  • Restricted: n! / (r!(n-r)!) × restrictions
Applications:
  • Probability: Event combinations
  • Statistics: Sample combinations
  • Computer Science: Algorithm analysis
  • Cryptography: Key combinations
  • Games: Card combinations
What Makes This Calculator Special?

Our combination calculator stands out with its comprehensive approach to combinatorial analysis and educational value:

Multiple Types

Supports standard, repetition, multiset, and restricted combinations with specialized algorithms for each type.

Educational Focus

Step-by-step solutions help students understand the mathematical process behind combination calculations.

Visual Analysis

Interactive charts and visualizations help understand combination patterns and relationships.

Combination Calculator

What is Combination Calculator?

This advanced combination calculator provides comprehensive solutions for calculating combinations, permutations, and factorial with step-by-step explanations, making it essential for combinatorics, probability, and statistics.

This tool is designed to handle complex combination calculations with precision and clarity, providing both numerical results and educational insights.

How is it Calculated?

Combinations are calculated using different formulas depending on the type:

Standard Combination (nCr):

C(n,r) = n! / (r!(n-r)!)

Where n is the total number of items and r is the number of items to select.

With Repetition:

C(n+r-1,r) = (n+r-1)! / (r!(n-1)!)

When items can be repeated in the selection.

Why is it Important?

Accurate combination calculations are critically important in many aspects:

  • Accuracy: Precise calculations are essential for reliable results
  • Understanding: Step-by-step solutions help learn mathematical concepts
  • Verification: Double-check manual calculations and verify results
  • Efficiency: Save time on complex mathematical operations
How to Use the Tool?

Using the Combination Calculator is very simple:

  1. Enter Values: Input the total number of items (n) and items to select (r)
  2. Select Type: Choose the type of combination calculation needed
  3. Set Method: Choose calculation approach (formula, step-by-step, or recursive)
  4. Click Calculate: View results instantly with step-by-step solutions
Calculation Example

Let's work through some sample combination calculations:

Example 1: Calculate 5C3 (5 items, select 3)
Given Values:
n = 5 (total items)
r = 3 (items to select)
Formula:
C(n,r) = n! / (r!(n-r)!)
Step-by-Step Solution:
Step 1: C(5,3) = 5! / (3!(5-3)!)
Step 2: C(5,3) = 5! / (3! × 2!)
Step 3: C(5,3) = 120 / (6 × 2) = 10
Answer: 5C3 = 10
Example 2: Calculate 7C4 (7 items, select 4)
Given Values:
n = 7 (total items)
r = 4 (items to select)
Formula:
C(n,r) = n! / (r!(n-r)!)
Step-by-Step Solution:
Step 1: C(7,4) = 7! / (4!(7-4)!)
Step 2: C(7,4) = 7! / (4! × 3!)
Step 3: C(7,4) = 5040 / (24 × 6) = 35
Answer: 7C4 = 35
Key Features
Calculation Features:
  • Precise Results: High-accuracy combination calculations
  • Step-by-Step Solutions: Detailed calculation process
  • Multiple Types: Standard, repetition, multiset, and restricted
  • Error Handling: Comprehensive input validation
Educational Features:
  • Learning Support: Educational explanations and examples
  • Formula Display: Mathematical formulas and concepts
  • Practice Examples: Sample problems and solutions
  • Interactive Interface: User-friendly design for all skill levels
Common Use Cases
Academic Applications:
  • Mathematics: Combinatorics and probability problems
  • Statistics: Sample combinations and selections
  • Computer Science: Algorithm analysis and optimization
  • Physics: Quantum mechanics and statistical mechanics
Real-World Applications:
  • Cryptography: Key combinations and security
  • Games: Card combinations and puzzle solving
  • Business: Team selection and resource allocation
  • Sports: Tournament combinations and matchups
Tips and Best Practices
Input Guidelines:
  • Valid Range: n and r should be positive integers
  • Logical Constraints: r should not exceed n
  • Large Numbers: Be cautious with very large values
  • Precision: Choose appropriate calculation method
Understanding Results:
  • Interpretation: Results represent possible combinations
  • Verification: Double-check with manual calculations
  • Context: Consider the specific problem context
  • Limitations: Understand when combinations apply

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