Binomial Coefficient Calculator
Calculate binomial coefficients (n choose r) with step-by-step solutions and Pascal's triangle
🔢 Binomial Coefficient Calculator
Calculate: C(5, 2) = "n choose r"
Formula
C(n,r) = n! / (r! × (n-r)!)
Where:
- C(n,r) =Binomial coefficient 'n choose r'
- n =Total number of items
- r =Number of items to choose
- n! =n factorial (n × (n-1) × ... × 1)
- (a+b)^n =Binomial expansion using coefficients
About Binomial Coefficients
Binomial coefficients represent the number of ways to choose r items from n items, fundamental in combinatorics and probability theory.
Key Applications:
- • Combinatorics: Counting combinations and arrangements
- • Probability: Binomial probability distributions
- • Algebra: Binomial theorem expansions
- • Pascal's Triangle: Number patterns and properties
Pascal's Triangle Properties:
- • Each number is the sum of the two above it
- • Row n contains coefficients for (a+b)^n
- • Sum of row n equals 2^n
- • Symmetric: C(n,r) = C(n,n-r)
Real-World Examples:
- • Lottery combinations: C(49,6) = 13,983,816
- • Team selection: Choose 5 players from 12
- • Genetics: Probability of trait combinations
- • Statistics: Binomial test calculations
Formula Variations:
- • C(n,r) = n! / (r! × (n-r)!)
- • C(n,r) = P(n,r) / r!
- • C(n,r) = C(n-1,r-1) + C(n-1,r)
Tips
- Enter the dimensions in the same unit for consistency.
- Results update automatically as you type.
- Use the unit selector to convert between different measurement systems.