Binomial Theorem ○◂ Entries

↩ 🔵 🔵 🔵

---

Primary

၊၊||၊|။

Binomial Theorem ○◂｜Definition｜1st｜20260207190909-00-⌔

Binomial theorem - Wikipedia

Binomial theorem

The binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ appears as the k th entry in the n th row of Pascal’s triangle (where the top is the 0th row $\left(\genfrac{}{}{0ex}{}{0}{0}\right)$). Each entry is the sum of the two above it.

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ⁠ $(x+y{)}^n$ ⁠ expands into a polynomial with terms of the form ⁠ $a{x}^k{y}^m$ ⁠, where the exponents ⁠ $k$ ⁠ and ⁠ $m$ ⁠ are nonnegative integers satisfying ⁠ $k+m=n$ ⁠ and the coefficient ⁠ $a$ ⁠ of each term is a specific positive integer depending on ⁠ $n$ ⁠ and ⁠ $k$ ⁠. For example, for ⁠ $n=4$ ⁠,

$(x+y{)}^4=x^4+4x^{3}y+6x^2{y}^2+4x{y}^3+y^4.$

The coefficient ⁠ $a$ ⁠ in each term ⁠ $a{x}^k{y}^m$ ⁠ is known as the binomial coefficient ⁠ $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ ⁠ or ⁠ $\left(\genfrac{}{}{0ex}{}{n}{m}\right)$ ⁠ (the two have the same value). These coefficients for varying ⁠ $n$ ⁠ and ⁠ $k$ ⁠ can be arranged to form Pascal’s triangle¹. These numbers also occur in combinatorics, where ⁠ $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ ⁠ gives the number of different combinations (i.e. subsets) of ⁠ $k$ ⁠ elements that can be chosen from an ⁠ $n$ ⁠ -element set. Therefore ⁠ $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ ⁠ is usually pronounced as ”⁠ $n$ ⁠ choose ⁠ $k$ ⁠”.

Printed 2026-06-28.

(echo:: @ ᯤ)

Footnotes

1. “Pascal’s triangle | Definition & Facts | Britannica”. Encyclopedia Britannica. Archived from the original on 2025-09-29. Retrieved 2026-03-25. ↩

Link to original

Secondary

• • •

---