All calculatorsBinomial Coefficient Calculator
Compute combinations C(n, k) = “n choose k” exactly with fast BigInt arithmetic. See steps, an n–k–(n−k) mini chart, a size magnitude gauge, and (optional) binomial probability P(X = k).
Background
The binomial coefficient counts the number of ways to choose k items from n without order: . We compute it exactly via a reduced multiplicative product to avoid huge factorials.
How this calculator works
- We compute C(n,k) exactly using a reduced product with per-step GCD cancellations (no giant factorials).
- Magnitude is shown as log₁₀ C and scientific notation if very large.
- If enabled, we also compute P(X=k) = C(n,k)p^k(1−p)^{n−k} for the binomial distribution.
Formula & Equation Used
Binomial coefficient (definition):
Multiplicative form:
Binomial probability (optional):
Example Problems & Step-by-Step Solutions
Example 1 — Poker hands: C(52, 5)
Using the reduced product with cancellations gives the exact integer: 2,598,960.
Example 2 — Symmetry: C(10, 3) = C(10, 7)
Compute the shorter side (k = min(k, n−k) = 3) for speed; result: 120.
Example 3 — Binomial probability with p = 0.5
For n = 30, k = 15, p = 0.5, we get P(X=15) ≈ 0.144 (via log-space).
Frequently Asked Questions
Q: How large can n be?
This tool computes exact integers with BigInt and is comfortable up to n≈500 (often beyond) thanks to GCD reductions.
Q: Why not use factorials directly?
Factorials explode in size. The reduced product with cancellations keeps numbers manageable and exact.
Q: Is the result rounded?
No. The main result is an exact integer. For readability, we also show scientific notation and log₁₀ magnitude.
