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Binomial Expansion and Factorial Notation
The binomial theorem describes the algebraic expansion of powers of a binomial.
Learning Objective

Use factorial notation and Pascal's Triangle to find the coefficients of a binomial expansion
Key Points

According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form a^{x}b^{y}c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b.

The factorial of a nonnegative integer n, denoted by n!, is the product of all positive integers less than or equal to n.

Binomial coefficients can be written as
$_{n}{C}_{k}$ and are defined in terms of the factorial function n!.
Term

factorial
The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 * 2 * 3 * 4 * 5 = 120.
Full Text
The binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)^{n} into a sum involving terms of the form a^{x}b^{y}c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.
In mathematics, the factorial of a nonnegative integer n, denoted by n!, is the product of all positive integers less than or equal to n . For example,
The value of 0! is 1, according to the convention for an empty product. The factorial operation is encountered in many different areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects). The definition of the factorial function can also be extended to noninteger arguments, while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis.
The coefficients that appear in the binomial expansion are called binomial coefficients.
These are usually written
The coefficient of x^{n−k}y^{k} is given by the formula:
Note that, although this formula involves a fraction, the binomial coefficient
Key Term Reference
 binomial
 Appears in this related concepts: Sums, Differences, Products, and Quotients, Factoring Trinomials of the Form ax^2 + bx + c; Perfect Squares, and Special Factorizations and Binomials
 binomial coefficient
 Appears in this related concepts: Combinations and Binomial Expansions and Pascal's Triangle
 coefficient
 Appears in this related concepts: Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1, Balancing Chemical Equations, and Polynomials: Introduction, Addition, and Subtraction
 exponent
 Appears in this related concepts: Logarithms of Products, Simplifying Expressions of the Form log_a a^x and a(log_a x), and Logarithms of Powers
 fraction
 Appears in this related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 function
 Appears in this related concepts: Solving Differential Equations, Four Ways to Represent a Function, and Modal Mixture
 integer
 Appears in this related concepts: Scientific Notation, Total Number of Subsets, and Finding a Specific Term
 operation
 Appears in this related concepts: Designing the Operation, Integer Exponents, and Converting between Exponential and Logarithmic Equations
 permutation
 Appears in this related concepts: Permutations, Permutations: Notation; n Objects Taken k at a Time, and Permutations of Nondistinguishable Objects
 sequence
 Appears in this related concepts: Summing an Infinite Series, Introduction to Sequences, and Finding the General Term
 set
 Appears in this related concepts: Sequences of Statements, Sequences, and Expressions and Sets of Numbers
 term
 Appears in this related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
 zero
 Appears in this related concepts: Zeroes of Linear Functions, Finding Zeroes of Factored Polynomials, and Reducing Equations to a Quadratic
Sources
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Cite This Source
Source: Boundless. “Binomial Expansion and Factorial Notation.” Boundless Algebra. Boundless, 03 Jul. 2014. Retrieved 21 May. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/sequencesseriesandcombinatorics8/thebinomialtheorem58/binomialexpansionandfactorialnotation24311213/