Binomial Coefficient Calculator
Calculate combinations (n choose k) using the binomial coefficient formula
Calculate Binomial Coefficient
The total number of items in the set (0 ≤ n ≤ 1000)
The number of items to choose from the set (0 ≤ k ≤ n)
Binomial Coefficient Result
Formula: C(n,k) = n! / (k! × (n-k)!)
Symmetry property: C(0,0) = C(0,0)
Pascal's Triangle Row 0
The highlighted value is C(0,0) = 1
Step-by-Step Solution
Step 1: Formula
The binomial coefficient formula is:
Step 2: Special Case
When k = 0, C(n,0) = 1 (there is exactly one way to choose 0 items)
Common Examples
Quick Reference
Calculator Tips
Order doesn't matter in combinations
k cannot be greater than n
Both n and k must be non-negative integers
Use symmetry for large values of k
Applications
Understanding Binomial Coefficients
What are Binomial Coefficients?
Binomial coefficients, denoted as C(n,k) or "n choose k," represent the number of ways to choose k items from a set of n items where the order of selection doesn't matter. They are fundamental in combinatorics, probability theory, and algebra.
Key Properties
- •Symmetry: C(n,k) = C(n,n-k)
- •Edge cases: C(n,0) = C(n,n) = 1
- •Pascal's identity: C(n,k) = C(n-1,k-1) + C(n-1,k)
- •Sum: ∑C(n,k) = 2ⁿ for k=0 to n
Combinations vs. Permutations
Real-World Examples
Binomial Theorem Connection
Binomial coefficients appear in the binomial theorem, which describes the expansion of powers of binomials:
For example: (x + y)³ = C(3,0)x³ + C(3,1)x²y + C(3,2)xy² + C(3,3)y³ = x³ + 3x²y + 3xy² + y³