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 $(x+y)^4$
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:
Key Term Reference
 binomial
 Appears in these related concepts: Multiplying Algebraic Expressions, Factoring a Difference of Squares, and Factoring Perfect Square Trinomials
 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
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