Binomial Coefficient Calculator

Combinations Calculator

Calculate binomial coefficients (combinations) with detailed step-by-step solutions and Pascal's triangle visualization.

Input Values

Non-negative integer (max 170)

Must be ≤ n

Binomial Coefficient Formula:
C(n,k) = n! / (k! × (n-k)!)
Also written as (n choose k) or ⁿCₖ

What is a Binomial Coefficient?

The binomial coefficient C(n,k) represents the number of ways to choose k items from n items without regard to order.

Formula:

C(n,k) = n! / (k! × (n-k)!)

Example:

C(5,2) = 5!/(2!×3!) = 120/(2×6) = 10

10 ways to choose 2 items from 5

Features

  • Step-by-step calculation
  • Factorial computation
  • Pascal's triangle visualization
  • Symmetry property display
  • Real-world interpretation
  • Large number support (up to n=170)
  • Copy results
  • Quick examples
  • Input validation

Properties

Symmetry:

C(n,k) = C(n,n-k)

Pascal's Identity:

C(n,k) = C(n-1,k-1) + C(n-1,k)

Sum:

Σ C(n,k) = 2ⁿ (for k=0 to n)

Special Cases:

C(n,0) = C(n,n) = 1

Common Examples

Lottery (6/49):

C(49,6) = 13,983,816 combinations

Poker Hand:

C(52,5) = 2,598,960 hands

Committee:

C(10,3) = 120 ways to choose 3 from 10

Binary Strings:

C(8,3) = 56 strings with 3 ones

Use Cases

  • Probability and statistics
  • Combinatorics problems
  • Lottery and gambling odds
  • Committee selection
  • Binomial theorem expansion
  • Graph theory
  • Computer science algorithms
  • Genetics and biology

Applications

Real-World Applications:

Probability:

Calculate odds in games and lotteries

Statistics:

Binomial distribution calculations

Computer Science:

Algorithm analysis and complexity

Genetics:

Calculate genetic combination probabilities

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