binomial coefficient

二项式系数

Combinatorics · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: n choose k

Translation evidence: Literature usage

Meaning in context

For integers n ≥ 0 and 0 ≤ k ≤ n, the binomial coefficient counts the k-element subsets of an n-element set and equals n!/(k!(n−k)!). It is defined as zero for integer k outside that range.

对整数 n≥0，二项式系数表示 n 元集合的 k 元子集数；0≤k≤n 时为 n!/(k!(n−k)!)，范围外的整数 k 对应零。

Example

$\binom{6}{2}=15$.

Usage and distinctions

Subset order does not matter; ordered selections have an additional factor k!.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Joy Morris — Combinatorics, version 2.1.1Section 3.2, Theorem 3.2.3 and Notation 3.2.4 · Checked 2026-10-14

Translation sources

1. 上海交通大学 — CS0901 组合数学（2026）§1.1 十二重计数法，第3种情形「二项式系数」 · Checked 2026-10-14

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