Binomial Coefficient Calculator
Compute combinations quickly using n choose k formula, factorials, and symmetry, with instant equation preview, examples, and accurate big-integer results.
Inputs
Equation Preview
Helping Notes
Use integers for n and k. The value is C(n,k) = n! / (k!(n−k)!). Symmetry: C(n,k) = C(n,n−k).
Large answers are exact using big integers; an additional scientific-notation approximation is shown for readability.
Results
Inputs Summary
Binomial Coefficient
Approximation
What Is a Binomial Coefficient Calculator?
A Binomial Coefficient Calculator evaluates combinations, usually written as , which count the ways to choose items from without order. It returns exact integers, displays algebraic identities, and provides optional decimal or scientific formats for very large results. Typical uses include counting problems, lottery odds, subsets, sampling without replacement, coefficients in polynomial expansions, and probability mass functions for the binomial distribution.
About the Binomial Coefficient Calculator
The core definition is , with symmetry and the Pascal identity . For numerical stability and speed, the calculator can use the multiplicative form (with ) to avoid intermediate overflow. It also reports useful bounds like and when quick estimates are helpful. For factor‑exact outputs, a prime‑factor method based on Legendre’s formula constructs the integer precisely.
Definition:
Symmetry: Pascal:
Multiplicative:
Bounds:
How to Use This Binomial Coefficient Calculator
- Enter non‑negative integers and . The tool auto‑reduces to for efficiency.
- Choose output: exact integer, scientific notation, or show steps (factorial, multiplicative, recursive).
- Press calculate to see the value, symmetry use (if applied), and optional bounds/estimates for context.
- Copy the result and the derivation for assignments, proofs, or probability calculations.
Examples
- Small values: , .
- Using symmetry: .
- Probability link: In , .
- Binomial expansion: Coefficient of in is .
- Multiplicative steps: .
Formula Snippets Ready for Rendering
FAQs
What does \u201cn choose k\u201d represent?
It counts the number of size‑ subsets of an -element set; order does not matter.
Why use the multiplicative form?
It avoids enormous factorials, reduces cancellation early, and is faster and safer for large .
What if k > n?
The value is zero by convention; the calculator enforces domain checks and reports invalid inputs clearly.
How large can n be?
Arbitrary‑precision integers are supported in exact mode; scientific notation summarises extremely large outputs.
Why is symmetry helpful?
Because computing minimizes steps and improves numerical stability.
How does this relate to the binomial theorem?
They are the coefficients in .
Can I compute values modulo m?
Use prime‑factor or modular methods; results can be reduced mod after the exact computation.