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

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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Summary

The Binomial Coefficient Calculator computes C(n,k) — read "n choose k" — which counts the number of ways to select k items from a set of n distinct items when order does not matter. It uses the formula C(n,k) = n! / (k! × (n−k)!) and handles large values without rounding errors. The tool also displays the full Pascal's Triangle row for your chosen n so you can see all coefficients at once.

How it works

Enter the total number of items (n) in the first field.
Enter the number of items to choose (k) in the second field.
The result C(n,k) is computed instantly using exact integer arithmetic.
The step-by-step formula breakdown shows how the factorials are applied.
The Pascal's Triangle row for n is displayed, with C(n,k) highlighted.

Use cases

Calculating lottery odds and probability problems.
Expanding binomial expressions such as (a+b)^n using binomial theorem.
Counting combinations in statistics and probability courses.
Computing sample sizes and hypergeometric distribution values.
Verifying Pascal's Triangle entries for combinatorics homework.
Solving card game and board game probability questions.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu