# Sequences, Probability, and Counting Theory

## Binomial Theorem


### Learning Objectives

1. Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)


### Objective 1: Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)

Pascal’s triangle helps us find the coefficients of the terms in the expansion of a binomial.

To find the coefficients of the terms, we write our expansion again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients. The array to the right is called Pascal’s Triangle.

Notice that in each expansion the powers of a in each term decrease from n to 0, and the powers of b increase from 0 to n.

Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.

To find the coefficients of the expansion of the binomial  , go to the row that has the value n as a second entry.


### Practice Makes Perfect


Use Pascal’s Triangle to expand a binomial.


A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find 
  without multiplying the binomial by itself 
  times.

### Identifying Binomial Coefficients

In Counting Principles, we studied combinations. In the shortcut to finding 
  we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation 
  instead of 
  but it can be calculated in the same way. So
The combination 
  is called a binomial coefficient. An example of a binomial coefficient is 
 


### Using the Binomial Theorem

When we expand 
  by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand 
  we might multiply 
  by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.
First, let’s examine the exponents. With each successive term, the exponent for 
  decreases and the exponent for 
  increases. The sum of the two exponents is 
  for each term.
Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.
Another way to see the coefficients is to examine the expansion of a binomial in general form, 
  to successive powers 1, 2, 3, and 4.
Can you guess the next expansion for the binomial 
  
See , which illustrates the following:
1. There are 
  terms in the expansion of 
  
2. The degree (or sum of the exponents) for each term is 
  
3. The powers on 
  begin with 
  and decrease to 0.
4. The powers on 
  begin with 0 and increase to 
  
5. The coefficients are symmetric.
To determine the expansion on 
  we see 
  thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of 
  the pattern is as follows:
1. Introduce 
  and then for each successive term reduce the exponent on 
  by 1 until 
  is reached.
2. Introduce 
  and then increase the exponent on 
  by 1 until 
  is reached.


The next expansion would be
But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal's Triangle, shown in . Pascal didn't invent the triangle. The underlying principles had been developed and written about for over 1500 years, first by the Indian mathematician (and poet) Pingala in the second century BCE. Others throughout Asia and Europe worked with the concepts throughout, and the triangle was first published in its graphical form by Omar Khayyam, an Iranian mathematician and astronomer, for whom the triangle is named in Iran. French mathematician Blaise Pascal repopularized it when he republished it and used it to solve a number of probability problems.
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3rd row, flank the ends of the rows with 1’s, and add 
  to find the middle number, 2. In the 
  row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.
To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.

### Using the Binomial Theorem to Find a Single Term

Expanding a binomial with a high exponent such as 
  can be a lengthy process.
Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.
Note the pattern of coefficients in the expansion of 
  
The second term is 
  The third term is 
  We can generalize this result.

### Key Equations


### Key Concepts

1.  
  is called a binomial coefficient and is equal to 
  See .
2. The Binomial Theorem allows us to expand binomials without multiplying. See .
3. We can find a given term of a binomial expansion without fully expanding the binomial. See .

### Section Exercises


### Verbal 


### Algebraic

For the following exercises, evaluate the binomial coefficient.
For the following exercises, use the Binomial Theorem to expand each binomial.
For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.
For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

### Graphical

For the following exercises, use the Binomial Theorem to expand the binomial 
  Then find and graph each indicated sum on one set of axes.

### Extensions
