Binomial Coefficient Calculator (n choose k)

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

In short

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

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.

Complete the required fields to see the result.

What this calculator does

Exact combinations C(n, k) and permutations P(n, k) for n up to 1,000. Formula, worked example, sources, FAQs and limitations included.

You enter total items n and chosen items k, and the binomial coefficient calculator (n choose k) shows the answer along with the working, so you can check it or try other numbers.

The page shows the formula, a numeric worked example, and the assumptions that affect this math calculation.

Inputs and what they mean

Total items n: — number value.
Chosen items k: — number value.

How to use it

Enter each value in the unit shown next to the box (use the unit converter first if your numbers are in other units).
Check the breakdown to see every intermediate step.
Read the limitations before relying on the result.

Formula

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.

Worked example

Choose 3 from 10

10! ÷ (3! × 7!) = 720 ÷ 6 = 120.

Reading the result

The headline figure is the main answer. Any breakdown underneath shows the parts that make it up, so you can check the working and see what changes when you adjust an input.

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.

Reference: OpenStax

Last reviewed: November 3, 2026

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.

Sources

Method based on OpenStax.

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