# Binomial Coefficient Calculator (n choose k)

> Exact combinations C(n, k) and permutations P(n, k) for n up to 1,000.

Page: https://freecalculator.tech/tools/binomial-coefficient-calculator
Last reviewed: 2026-11-03

## Formula

C(n, k) = n! ÷ (k! (n − k)!), the coefficient of x^k in (1 + x)^n. Permutations P(n, k) = n! ÷ (n − k)! count ordered selections.

- Source: OpenStax College Algebra 2e §9.6, Binomial Theorem.
- Computed with exact integer arithmetic — no rounding.
- C(n, k) = n! ÷ (k! (n − k)!), the coefficient of x^k in (1 + x)^n. Permutations P(n, k) = n! ÷ (n − k)! count ordered selections.
- Inputs used: Total items n, Chosen items k.

## Inputs

- Total items n
- Chosen items k

## Worked example: Choose 3 from 10

C(10, 3) = 120; P(10, 3) = 720.
1. 10! ÷ (3! × 7!) = 720 ÷ 6 = 120.

## Common questions

### What is the difference between combinations and permutations?

Combinations ignore order; permutations count each ordering separately.

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

1 — there is exactly one way to choose nothing.

### How does this relate to the binomial theorem?

(a + b)^n = Σ C(n, k) a^(n−k) b^k, so C(n, k) are the coefficients of the expansion.

### How do I use the Binomial Coefficient Calculator (n choose k)?

Enter the required values for Total items n, Chosen items k. The calculator applies the formula on this page and shows the main result with any supporting breakdown so you can check the arithmetic.

### What formula does the Binomial Coefficient Calculator (n choose k) use?

C(n, k) = n! ÷ (k! (n − k)!), the coefficient of x^k in (1 + x)^n. Permutations P(n, k) = n! ÷ (n − k)! count ordered selections. The visible formula section above lists the calculation path and the edge cases the page handles, so the result can be checked without relying on the form alone.

### Can the Binomial Coefficient Calculator (n choose k) be used for exact decisions?

Use it as a calculation aid, not as a substitute for checking the underlying rule, contract, policy, or professional advice that applies to your situation. When a result depends on local rules, personal details, prices, or dates, enter those values yourself and confirm them before acting.

## Limitations and assumptions

- n up to 1,000.
- Without-repetition selections only.
- Exact mathematical relationships, rounded for display. Check units and rounding before using a result in graded, engineering or financial work.
- The result depends on the values you enter for this binomial coefficient calculator (n choose k); it does not supply missing rates, rules, prices, dates, or assumptions for you.

## Sources

- OpenStax — https://openstax.org/books/college-algebra-2e/pages/9-6-binomial-theorem

## Cite this page

- APA: Free Calculator. (2026, November 3). Binomial Coefficient Calculator (n choose k). https://freecalculator.tech/tools/binomial-coefficient-calculator
- MLA: "Binomial Coefficient Calculator (n choose k)." Free Calculator, 3 Nov. 2026, https://freecalculator.tech/tools/binomial-coefficient-calculator.
- Link: Binomial Coefficient Calculator (n choose k) — Free Calculator, https://freecalculator.tech/tools/binomial-coefficient-calculator