What is the difference between C(n,k) and P(n,k)?

C(n,k) counts selections where order does not matter (combinations), while P(n,k) counts arrangements where order matters (permutations). C(n,k) = P(n,k) / k!.

What is C(n,0) and C(n,n)?

Both equal 1. There is exactly one way to choose nothing from a set, and exactly one way to choose everything from a set.

Yes. C(n,0) = 1 for any n ≥ 0, and C(0,0) = 1. k must be between 0 and n inclusive; values outside that range yield 0 by convention.

How does this handle very large numbers?

The calculator uses JavaScript's BigInt to perform exact integer arithmetic, so results are precise even for large inputs like C(100,50).

What is Pascal's Triangle and why is it shown?

Pascal's Triangle lists all binomial coefficients C(n,0) through C(n,n) for each row n. Each entry is the sum of the two entries above it, which makes it a useful visual reference for combinatorics.

Binomial Coefficient Calculator

Name: Binomial Coefficient Calculator
Availability: InStock
Author: Nham Vu

Enter n and k to instantly compute C(n,k) — the binomial coefficient — with the full formula and Pascal's Triangle row.

Enter Values

n — Total items

k — Items chosen

Quick Examples

Result

Enter n and k above, then click Calculate.

Pascal's Triangle — Row n

All C(n,k) values for your chosen n. The highlighted cell is your result.

Awaiting calculation.

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