# Binomial Coefficient Calculator

Calculate n choose k (binomial coefficients) instantly. Shows the full factorial formula, step-by-step work, and result. Free online combinatorics tool.

## What this calculates

Calculate any binomial coefficient C(n, k), also called "n choose k." Enter your values of n and k to get the result with the full factorial expansion shown.

## Inputs

- **n (total items)** — min 0 — The total number of items. Must be a non-negative integer.
- **k (items to choose)** — min 0 — The number of items to choose. Must satisfy 0 <= k <= n.

## Outputs

- **C(n, k)** — The binomial coefficient, also written as 'n choose k'.
- **Formula Expansion** — formatted as text — The factorial formula used in the calculation.
- **n!** — The factorial of n.
- **k!** — The factorial of k.
- **(n - k)!** — The factorial of (n - k).

## Details

The binomial coefficient C(n, k) counts the number of ways to choose k items from a set of n items, where order does not matter. It shows up everywhere in math, from probability to algebra.

**The Formula:**

C(n, k) = n! / (k! × (n - k)!)

Where n! (n factorial) means n × (n-1) × (n-2) × ... × 2 × 1.

**Quick Example:**

How many ways can you pick 3 students from a class of 10?

C(10, 3) = 10! / (3! × 7!) = 720 / 6 = 120 ways.

**Key Properties:**

- **Symmetry:** C(n, k) = C(n, n-k). Choosing 3 items from 10 is the same count as choosing 7 items from 10.
- **Pascal's Rule:** C(n, k) = C(n-1, k-1) + C(n-1, k). Each entry in Pascal's triangle is the sum of the two entries above it.
- **Sum of a row:** C(n, 0) + C(n, 1) + ... + C(n, n) = 2^n.

**Where You See Binomial Coefficients:**

- Probability (how many ways an event can happen)
- The Binomial Theorem: (a + b)^n = sum of C(n, k) × a^(n-k) × b^k
- Pascal's Triangle
- Counting problems in combinatorics

## Frequently Asked Questions

**Q: What is the difference between combinations and permutations?**

A: Combinations (C(n,k)) count selections where order does not matter. Permutations (P(n,k)) count arrangements where order matters. For example, choosing 3 people from 5 gives C(5,3) = 10 combinations, but P(5,3) = 60 permutations, because each group of 3 can be arranged in 3! = 6 different orders.

**Q: Why does C(n, k) equal C(n, n-k)?**

A: Choosing which k items to include is the same as choosing which n-k items to exclude. For example, picking 2 cards from 5 means leaving 3 behind, so C(5,2) = C(5,3) = 10. This symmetry is visible in Pascal's triangle, where each row reads the same forwards and backwards.

**Q: How does the binomial coefficient relate to Pascal's triangle?**

A: Pascal's triangle is built entirely from binomial coefficients. Row n contains C(n,0), C(n,1), ..., C(n,n). Each entry is the sum of the two entries directly above it, which is Pascal's Rule: C(n,k) = C(n-1,k-1) + C(n-1,k).

**Q: What is the largest binomial coefficient for a given n?**

A: The largest value in row n of Pascal's triangle is C(n, floor(n/2)). For even n, there is one maximum in the center. For odd n, two adjacent entries tie for the largest. For example, in row 6 the maximum is C(6,3) = 20.

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Source: https://vastcalc.com/calculators/math/binomial-coefficient
Category: Math
Last updated: 2026-04-08