The Binomial Formula

Hello. This is a math post about the binomial formula. The binomial formula algebra technique allows us to expand binomials such as (a + b)^{n} where n \geq 2.


Table of Contents

  1. Factorials
  2. Binomial Coefficient
  3. Pascal’s Triangle
  4. The Binomial Formula
  5. Examples


Factorials

Before we get right into the binomial formula, we need the ingredients first. We start with the factorial.

Suppose we have something like 3 \cdot 2 \cdot 1 = 6 and 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120. We would like a more compact form of writing this (especially for larger numbers).

The factorial form is expressed as n! where n \geq 1 and:

\displaystyle n! = n \cdot (n - 1) \cdot (n - 2) \text{ ... } \cdot 1
The alternative formula is:

\displaystyle n! = \prod_{k = 1}^{n} k = 1 \cdot 2 \cdot \text{ ... } \cdot (n-1) \cdot n
We can express 3 \cdot 2 \cdot 1 = 6 as 3! = 6 and 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 as 5! = 120.

Some Factorial Properties

n! = n \cdot (n -1)!
0! = 1


Binomial Coefficient

The next piece we need is the binomial coefficient. In terms of practical applications, this binomial coefficient represents the number of ways (combinations) of choosing k unordered items from n choices.

Mathematically, the binomial coefficient is represented as:

\binom{n}{k} = \dfrac{n!}{k! (n-k)!}
Some Properties

\binom{n}{n} = \binom{n}{0} = 1 for integers n \geq 0.
\binom{0}{0} = 1 (Corollary/Special Case from the above)


Pascal’s Triangle

We now look at Pascal’s Triangle. You may wonder what does this have to do with the binomial formula. As you read on, you will see. Below is the first 5 rows of Pascal’s Triangle.

\displaystyle \begin{array} {lcl} \begin{tabular}{rccccccccc}  & & & & & 1 & & & & \  & & & & 1 & & 1 & & & \  & & & 1 & & 2 & & 1 & & \  & & 1 & & 3 & & 3 & & 1 & \  & 1 & & 4 & & 6 & & 4 & & 1\  \end{tabular}  \end{array}

The numbers in Pascal’s triangle can also be expressed using binomial coefficients and the \binom{n}{k} notation as seen below.

PascalsTriangleCoefficient

The first 6 rows of Pascal’s traingle using binomial coefficients. Source: https://euclidlab.org/camp-euclid/2013/summer/group-1/index.php?title=Pascal%27s_Triangle_Problem

 

Note: The LaTeX (lay-tech) code that was used to generate the first Pascal triangle above was from \texttt{http://forum.mackichan.com/node/274}.


Binomial Formula

Now we have the necessary ingredients. Here is the binomial formula for n \geq 1

\begin{array} {lcl} \displaystyle (a + b)^{n} & = & \sum_{k = 0}^{n} \binom{n}{k} a^{n-k}b^{k} \ & = & \binom{n}{0}a^{n} + \binom{n}{1} a^{n-1}b + \text{ ...} + \binom{n}{n} b^{n} \ & = & a^{n} + na^{n-1}b + \text{ ...} + b^{n} \end{array}


Examples

Here are a few examples which apply the binomial formula.

1) Example One: (x + 2)^2

\displaystyle \begin{array} {lcl} (x + 2)^{2} & = & \sum_{k = 0}^{n = 2} \binom{2}{k} x^{2-k}2^{k}\ & = & \binom{2}{0} x^{2}2^{0} + \binom{2}{1} x^{1}2^{1} + \binom{2}{2} x^{0}2^{2}\ & = & 1\cdot x^2 + 2 \cdot 2x + 1 \cdot 4\ & = & x^2 + 4x + 4\ \end{array}
2) Example Two: (x - 5)^3

\displaystyle \begin{array} {lcl} (x - 5)^{3} & = & \sum_{k = 0}^{n = 3} \binom{3}{k} x^{3-k}(-5)^{k}\ & = & \binom{3}{0} x^{3}(-5)^{0} + \binom{3}{1} x^{2}(-5)^{1} + \binom{3}{2} x^{1}(-5)^{2} + \binom{3}{3} x^{0}(-5)^{3}\ & = & 1\cdot x^3 - 3 \cdot 5x^2 + 3 \cdot 25x - 1 \cdot 125\ & = & x^3 - 15x^2 + 75x - 125\ \end{array}
3) Example Three: (a - b)^4

\displaystyle \begin{array} {lcl} (a - b)^{4} & = & \sum_{k = 0}^{n = 4} \binom{4}{k} a^{4-k} (-b)^{k}\ & = & \binom{4}{0} a^{4}(-b)^{0} + \binom{4}{1} a^{3}(-b)^{1} + \binom{4}{2} a^{2}(-b)^{2} + \binom{4}{3} a^{1}(-b)^{3} + \binom{4}{4} a^{0}(-b)^{4}\ & = & 1 \cdot a^4 - 4 \cdot a^{3}b + 6 \cdot a^2 b^2 - 4ab^3 + 1 \cdot (-b)^{4}\ & = & a^4 - 4 a^{3}b + 6 a^2 b^2 - 4ab^3 + b^{4}\ \end{array}
Notice that with binomials with one positive term and one negative term the signs are alternating in the sum. This is seen in examples two and three.

The featured image is from http://exchangedownloads.smarttech.com/public/content/2b/2b616ad0-d9ab-496d-b82b-a2c2dc5dc381/previews/medium/0001.png.

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