5.6: Binomial Theorem
In this section, you will:
- Apply the Binomial Theorem.
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.
Finding the Number of Permutations of n Distinct Objects Using a Formula
For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of objects and we want to choose objects from the set in order, we write Another way to write this is a notation commonly seen on computers and calculators. To calculate we begin by finding the number of ways to line up all objects. We then divide by to cancel out the items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is Using factorials, we get the same result.
There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.
Note that the formula stills works if we are choosing all objects and placing them in order. In that case we would be dividing by or which we said earlier is equal to 1. So the number of permutations of objects taken at a time is or just
Formula for Permutations of n Distinct Objects
Given distinct objects, the number of ways to select objects from the set in order is
How To
Given a word problem, evaluate the possible permutations.
- Identify from the given information.
- Identify from the given information.
- Replace and in the formula with the given values.
- Evaluate.
Example 4
Finding the Number of Permutations Using the Formula
A professor is creating an exam of 9 questions from a test bank of 12 questions. How many ways can she select and arrange the questions?
- Answer
-
Substitute and into the permutation formula and simplify.
There are 79,833,600 possible permutations of exam questions!
Analysis
We can also use a calculator to find permutations. For this problem, we would enter 12, press the function, enter 9, and then press the equal sign. The function may be located under the MATH menu with probability commands.
Q&A
Could we have solved Example 4 using the Multiplication Principle?
Yes. We could have multiplied to find the same answer .
A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.
Try It #6
How many ways can the 7 actors line up?
Try It #7
How many ways can 5 of the 7 actors be chosen to line up?
Find the Number of Combinations Using the Formula
So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations . A selection of objects from a set of objects where the order does not matter can be written as Just as with permutations, can also be written as In this case, the general formula is as follows.
An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are ways to order 3 paintings. There are or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are not selecting 1 painting. There are 4 paintings we could choose not to select, so there are 4 ways to select 3 of the 4 paintings.
Formula for Combinations of n Distinct Objects
Given distinct objects, the number of ways to select objects from the set is
How To
Given a number of options, determine the possible number of combinations.
- Identify from the given information.
- Identify from the given information.
- Replace and in the formula with the given values.
- Evaluate.
Example 5
Finding the Number of Combinations Using the Formula
A fast food restaurant offers five side dish options. Your meal comes with two side dishes.
- ⓐ How many ways can you select your side dishes?
- ⓑ How many ways can you select 3 side dishes?
- Answer
-
-
ⓐ
We want to choose 2 side dishes from 5 options.
-
ⓑ
We want to choose 3 side dishes from 5 options.
-
ⓐ
We want to choose 2 side dishes from 5 options.
Analysis
We can also use a graphing calculator to find combinations. Enter 5, then press enter 3, and then press the equal sign. The function may be located under the MATH menu with probability commands.
Q&A
Is it a coincidence that parts (a) and (b) in Example 5 have the same answers?
No. When we choose r objects from n objects, we are not choosing objects. Therefore,
Try It #8
An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?
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
Binomial Coefficients
If and are integers greater than or equal to 0 with then the binomial coefficient is
Q&A
Is a binomial coefficient always a whole number?
Yes. Just as the number of combinations must always be a whole number, a binomial coefficient will always be a whole number.
Example 1
Finding Binomial Coefficients
Find each binomial coefficient.
- ⓐ
- ⓑ
- ⓒ
- Answer
-
Use the formula to calculate each binomial coefficient. You can also use the function on your calculator.
- ⓐ
- ⓑ
- ⓒ
Analysis
Notice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations.
Try It #1
Find each binomial coefficient.
- ⓐ
- ⓑ
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 Figure 1 , which illustrates the following:
- There are terms in the expansion of
- The degree (or sum of the exponents) for each term is
- The powers on begin with and decrease to 0.
- The powers on begin with 0 and increase to
- 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:
- Introduce and then for each successive term reduce the exponent on by 1 until is reached.
-
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 Figure 2 .
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3 rd 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.
The Binomial Theorem
The Binomial Theorem is a formula that can be used to expand any binomial.
How To
Given a binomial, write it in expanded form.
- Determine the value of according to the exponent.
- Evaluate the through using the Binomial Theorem formula.
- Simplify.
Example 2
Expanding a Binomial
Write in expanded form.
- ⓐ
- ⓑ
- Answer
-
-
ⓐ
Substitute
into the formula. Evaluate the
through
terms. Simplify.
-
ⓑ
Substitute
into the formula. Evaluate the
through
terms. Notice that
is in the place that was occupied by
and that
is in the place that was occupied by
So we substitute them. Simplify.
-
ⓐ
Substitute
into the formula. Evaluate the
through
terms. Simplify.
Analysis
Notice the alternating signs in part b. This happens because raised to odd powers is negative, but raised to even powers is positive. This will occur whenever the binomial contains a subtraction sign.
Try It #2
Write in expanded 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.
The (r+1)th Term of a Binomial Expansion
The term of the binomial expansion of is:
How To
Given a binomial, write a specific term without fully expanding.
- Determine the value of according to the exponent.
- Determine
- Determine
- Replace in the formula for the term of the binomial expansion.
Example 3
Writing a Given Term of a Binomial Expansion
Find the tenth term of without fully expanding the binomial.
- Answer
-
Because we are looking for the tenth term, we will use in our calculations.
Try It #3
Find the sixth term of without fully expanding the binomial.
Media
Access these online resources for additional instruction and practice with binomial expansion.