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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 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$ax^by^c$ , 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
$\displaystyle{\begin{pmatrix} n \\ k \end{pmatrix}}$ or$_{n}{C}_{k} $ and are defined in terms of the factorial function$n!$ .
Terms

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,
$5! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$ . 
binomial coefficient
A coefficient of any of the terms in the expansion of the binomial power
$(x+y)^n$
Full Text
Recall that the binomial theorem is an algebraic method of expanding a binomial that is raised to a certain power, such as
The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written
The coefficient of a term
Note that, although this formula involves a fraction, the binomial coefficient
In calculating coefficients, recall that the factorial of a nonnegative integer
Finally, you may recall that the factorial
Example
Use the binomial formula to find the expansion of
Start by substituting
In order to solve this, we will need to expand the summation for all values of
Recall that
Now we must evaluate each of the remaining combinations:
Substituting these integers into the expansion, we have:
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Key Term Reference
 binomial
 Appears in these related concepts: Multiplying Algebraic Expressions, Factoring Perfect Square Trinomials, and Factoring a Difference of Squares
 coefficient
 Appears in these related concepts: Factoring General Quadratics, Introduction to Variables, and Balancing Chemical Equations
 combination
 Appears in these related concepts: Composition of Functions and Decomposing a Function, Simplifying Exponential Expressions, and Combinations
 exponent
 Appears in these related concepts: Logarithms of Products, Logarithms of Powers, and Logarithms of Quotients
 fraction
 Appears in these related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 function
 Appears in these related concepts: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 integer
 Appears in these related concepts: Scientific Notation, Binomial Expansions and Pascal's Triangle, and Finding a Specific Term
 permutation
 Appears in these related concepts: Permutations, Permutations of Nondistinguishable Objects, and Permutations of Distinguishable Objects
 product
 Appears in these related concepts: Measuring Reaction Rates, Writing Chemical Equations, and Basic Operations
 sum
 Appears in these related concepts: Scientific Applications of Quadratic Functions, The Order of Operations, and What Are Polynomials?
 summation
 Appears in these related concepts: Applications and ProblemSolving, Interactions of Skeletal Muscles, and Series and Sigma Notation
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
Sources
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Cite This Source
Source: Boundless. “Binomial Expansion and Factorial Notation.” Boundless Algebra. Boundless, 14 Jun. 2016. Retrieved 24 Aug. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/combinatoricsandprobability343/thebinomialtheorem58/binomialexpansionandfactorialnotation24311213/