Use the 3 calculators below: calculate C(n,k) with step-by-step work, visualize row n of Pascal's triangle, or see the complete expansion of (a + b)n.
Binomial Coefficient Calculator
Calculation 1 — Binomial coefficient C(n, k)
Calculation 2 — Row n of Pascal's triangle
Calculation 3 — Binomial expansion (a + b)n
How to use the calculator
- Calculation 1: Enter n and k to get C(n,k) with the step-by-step calculation and its position in Pascal's triangle.
- Calculation 2: Enter n (up to 20) to see all rows of Pascal's triangle up to row n, with row n highlighted.
- Calculation 3: Enter n (up to 15) to see the full expansion of (a + b)n with all coefficients.
Binomial coefficient formula
| Notation | Formula | Example |
|---|---|---|
| C(n,k) or Cₙₖ | n! ÷ (k! × (n−k)!) | C(5,2) = 120 ÷ (2 × 6) = 10 |
| Symmetry property | C(n,k) = C(n, n−k) | C(5,2) = C(5,3) = 10 |
| Special cases | C(n,0) = C(n,n) = 1 | C(7,0) = 1 |
| Recursion (Pascal) | C(n,k) = C(n−1,k−1) + C(n−1,k) | C(5,2) = C(4,1) + C(4,2) |
Frequently asked questions
What is a binomial coefficient?
The binomial coefficient C(n,k), read "n choose k", counts the number of ways to choose k elements from a set of n without considering order. It is the same as combination: C(n,k) = Cₙₖ = n! ÷ (k! × (n−k)!).
Example: C(5,2) = 5!÷(2!×3!) = 120÷12 = 10. There are 10 ways to choose 2 objects from a group of 5.
How does the binomial coefficient relate to Pascal's triangle?
Each entry in Pascal's triangle at row n and column k (both starting at 0) equals C(n,k). The formation rule is: C(n,k) = C(n−1,k−1) + C(n−1,k) — each number is the sum of the two above it.
What is the difference between a binomial coefficient and a permutation?
Combination C(n,k): chooses k elements without regard to order. Permutation P(n,k) = n!÷(n−k)!: chooses k elements where order matters. C(n,k) = P(n,k) ÷ k!. Example: C(5,2)=10 combinations, P(5,2)=20 permutations.
What is the binomial theorem?
The binomial theorem states that (a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ, for k from 0 to n. The coefficients of each term are exactly the binomial coefficients C(n,k). Example: (a+b)² = C(2,0)·a² + C(2,1)·ab + C(2,2)·b² = a² + 2ab + b².
See also…
