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
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.
C(n,k) = n! / (k! × (n-k)!)
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
C(n,k) = C(n,n-k)
C(n,k) = C(n-1,k-1) + C(n-1,k)
Σ C(n,k) = 2ⁿ (for k=0 to n)
C(n,0) = C(n,n) = 1
Common Examples
C(49,6) = 13,983,816 combinations
C(52,5) = 2,598,960 hands
C(10,3) = 120 ways to choose 3 from 10
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:
Calculate odds in games and lotteries
Binomial distribution calculations
Algorithm analysis and complexity
Calculate genetic combination probabilities
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