In this explainer, we will learn how to use Pascal’s triangle to find the coefficients of the algebraic expansion of any binomial expression of the form .
We begin by considering the expansions of for consecutive powers of , starting with . Since any number raised to the power of zero equals 1 (note that we are using the convention that ),
Similarly, when , we have a somewhat trivial case:
However, for , things get a little more interesting. Using familiar algebra, we know
We now consider for this case. We will use our answer from to write the expansions as follows:
Expanding the parentheses, we have
Similarly, we can find the expansion of using the expansion of as follows:
We can now expand the parentheses to get
As you can see, if we were to try to find the expansion of this way, it could take a serious amount of time and effort. What we need is a better method that generalizes to higher powers. To find such a method, we will first look for patterns which might help us. We begin by organizing the expansions of one above the other to see if we can see any patterns.
Some of the most obvious patterns we notice are related to the diagonals: the coefficients of the terms in the first diagonal only contain ones, whereas the coefficients in the second diagonal contain consecutive integers.
Furthermore, we can see there is reflectional symmetry about the center.
Furthermore, we notice that on any given row, the sum of the indices equals . For example, on the row representing , the second term is . The index of is 3 and the index of is 1. Hence, their sum is equal to 4.
Finally, we see there is a relation between the coefficients on consecutive rows: if we add the two coefficients in the row above, we get the coefficient in the following row.
The triangle which makes up the binomial coefficients is generally referred to as Pascal’s triangle.
Definition: Pascal’s Triangle
Pascal’s triangle is a triangular array of the binomial coefficients. The rows are enumerated from the top such that the first row is numbered . Similarly, the elements of each row are enumerated from up to . The first eight rows of Pascal’s triangle are shown below.
Although, in much of the Western world, the triangle is named after the French mathematician Blaise Pascal, it was, in fact, well known to mathematicians centuries before in places such as China, Persia, and India.
Pascal’s triangle is easy to reproduce for small and is therefore extremely helpful in expanding binomials with moderate powers. Later, we will see how its properties give us a method to expand general binomials.
We need to be careful to differentiate between referring to rows using ordinals, such as first row and second row, and referring to them using the row number : when we say the second row, we are referring to the row for which .
Example 1: Using Pascal’s Triangle to Find Binomial Coefficients
Fady has been exploring the relationship between Pascal’s triangle and the binomial expansion. He has noticed that each row of Pascal’s triangle can be used to determine the coefficients of the binomial expansion of , as shown in the figure. For example, the fifth row of Pascal’s triangle can be used to determine the coefficients of the expansion of .
- By calculating the next row of Pascal’s triangle, find the coefficients of the expansion of .
- Fady now wants to calculate the coefficients for each of the terms of the expansion . By substituting into the expression above, or otherwise, calculate all of the coefficients of the expansion.
To calculate the seventh row of Pascal’s triangle, we start by writing out the sixth row. Then, since all rows start with the number 1, we can write this down. We can then add each consecutive pair of elements of the sixth row and write their sum in the gap beneath them. We will demonstrate this process below.
Starting with the first pair of terms, 1 and 5, we add them together to get 6 and place it into the space gap beneath them.
Moving onto the next pair of terms, we have , which we similarly add to the row.
Now we consider the middle terms .
Finally, we can use the symmetry of Pascal’s triangle to write the rest of the row.
Since the elements of Pascal’s triangle are the binomial coefficients, we can state that the coefficients of the terms of the expansion of will be 1, 6, 15, 20, 15, 6, and 1 respectively.
Since we can substitute for and write
Simplifying, we have
Therefore, the coefficients for each of the terms of the expansion are 16, 32, 24, 8, and 1.
Example 2: Using Pascal’s Triangle to Find Binomial Coefficients
Amer knows that he can use the 6th row of Pascal’s triangle to calculate the coefficients of the expansion .
- Calculate the numbers in the 6th row of Pascal’s triangle and, hence, write out the coefficients of the expansion .
- Now, by considering the different powers of and and using Pascal’s triangle, work out the coefficients of the expansion .
Recall that we can write out the rows of Pascal’s triangle by pairwise adding the terms in the previous rows. Therefore, starting from the first and second rows, which only contain ones, we can create the third rows by adding consecutive terms, as shown in the figure below.
Similarly, we can write the other rows using the same method, until we get to the sixth row.
Since the elements of Pascal’s triangle are the binomial coefficients, we can state that the coefficients of the terms of the expansion of will be 1, 5, 10, 10, 5, and 1 respectively.
To find the coefficients of the terms in the expansion of , we can first factor the 2 out of the parentheses as follows:
We can now substitute for in the expansion, to get
We can simplify this expression as follows:
Hence, the coefficients for each of the terms of the expansion are 32, , 320, , 160, and .
When working with binomial expansions we might only be interested in specific terms, or even the coefficients of a specific terms.
Individual terms can often be identified by looking for powers of a certain variable. Consider the powers of the variable in a simple expansion such as . The terms of the expansion will involve all of the integer powers of from to .
If however, both terms in a binomial contain the same variable, it might not be clear which powers of a certain variable will appear. One such example would be . Expressing we might observe that such an expansion would only contain even or odd powers of the variable from to .
Let’s take a look at an example of this.
Example 3: Using Pascals Triangle to Find the Coefficient in a Product of Binomial Expansions
Find the coefficient of in the expansion of .
Given that we have the product of two binomials raised to a power, it is usually helpful to expand each set of parentheses separately; then, we can consider their product. Since both binomials are raised to the third power, we can use find the coefficients for our terms using the third row of Pascal’s triangle.
Note that the single entry at the top of the triangle is conventionally referred to as the 0th row. To avoid ambiguity, we are using the row maked as .
Beginning with the first set of parentheses, we can apply the method for the expansion of a general binomial of power 3, which states that
Setting and , we have
Similarly, we can consider the second set of parentheses by substituting and into the equation we have:
Hence, we have
We can now consider which pairs of terms have a product containing . There are, in fact, only two pairs of such terms: and , and and :
Therefore, since both of these terms are going to contain , they will both contribute to the coefficient of this term in our final expansion. Hence, the final coefficient of will be the sum of their two respective coefficients. Since each has a coefficient of 3, the final coefficient of will be .
For larger values of , an increasing amount of work is required to find individual terms if our method involves writing out the entire expansion.
The remainder of this explainer will seek to provide some tricks reduce the amount of calculation required for such questions.
Example 4: Using Pascals Triangle to Find the Coefficient of a Specific Term in a Binomial Expansion
Find the coefficient of in the expansion of .
Recall that the elements of Pascal’s triangle give us the coefficients for the terms in a binomial expansion. Since we are expanding a binomial of power 8, we will use the 8th row of the triangle, .
Finding the coefficients from Pascal’s triangle is itself not enough to answer the question. Since we are trying to find the coefficient of in the final expansion we must also account the coefficients of terms within the binomial itself.
At this stage we could simplify using a substitution of and . This would allow us to lead to a more straight forward expansion of .
We might also choose to directly use the terms in the original binomial, and incorporate the coefficients found using Pascals triangle.
At this point we could simplify our terms and reach an answer, however for binomials with large powers, notice that writing out such an expansion can be quite laborious! It’s worth going through some tricks than can be used to save time.
Let’s consider which term (or terms) will contains . In our case we only need to pay attention to the powers of . As shown in the expansion, each term contains successively larger powers of .
Much like the corresponding row in Pascal’s triangle, we can label these terms using , where corresponds to the first term. The value of increases up to , where is the row number of Pascal’s triangle.
For this binomial expansion, is the only term relevant to finding the coefficient of . If the power of is larger or smaller than 5, the power of itself will also be larger or smaller than 5.
The th element in Pascals triangle gives us the coefficients labelled above.
Instead of writing out the entire expansion, this method gives us a shortcut to find individual terms.
Hence the coefficient of is 1 400 000.
As a final note, had the order of the terms in our previous binomial been reversed, would instead have appeared when . Since this means that we would have evaluated the term where .
Fortunately, this would not have changed our answer. The simplest way to understand why is to recognize that the th element and the th element in Pascal’s triangle are the same due symmetry. This means the coefficient (and hence our answer to the previous example) would remain unchanged.
The method show for finding individual terms is useful and can be generalised for binomials with large powers. One remaining flaw is that to find the th element of the th row in Pascal’s triangle, we still needed to construct rows of the triangle!
Although using Pascal’s triangle can seriously simplify finding binomial expansions for powers of up to around 10, much beyond this point it becomes impractical. It would, therefore, be helpful to see if there is a connection between consecutive elements in the rows of Pascal’s triangle.
As an example, let us consider the ninth row of Pascal’s triangle (i.e., the row labeled ). We consider the multipliers taking us from one element to the next. The figure represents this.
We can see clearly that there is a pattern linking one element to the next. In fact, we can express this in a general way as follows: to move from the th element to the th, we multiply by and divide by . This rule does not only apply to the ninth row but also generalizes to any row of Pascal’s triangle. Using this fact, we can expand binomials with arbitrarily large exponents.
Property: Connection between Consecutive Terms in the Same Row of Pascal’s Triangle
The connection between consecutive elements in the th row (which by convention we enumerate by ) in Pascal’s triangle is as follows: to move from the th element to the th, we multiply by .
The next couple of examples will demonstrate this fact.
Example 5: Using Pascal’s Triangle to Find Binomial Expansions
Write the first 5 terms of the expansion of in ascending powers of .
We will start by considering the coefficients of the first five terms of this expansion. The coefficients are given by the nineteenth row of Pascal’s triangle, that is, the row we label . The first element in any row of Pascal’s triangle is 1. Recall the connection between consecutive elements in a row in Pascal’s triangle: to move from the th element to the th, we multiply by . Applying this rule, we can calculate the 1st element by multiplying the 0th element by . Then, to find the second element, we multiply by . Continuing this way, we can find the first five terms in the row, as demonstrated in the figure below.
Therefore, the first five terms are given by
Simplifying, we have
Example 6: Using Pascal’s Triangle to Find Binomial Expansions
Fully expand the expression .
We will begin by finding the binomial coefficient. The coefficients are given by the eleventh row of Pascal’s triangle, which is the row we label . The first element in any row of Pascal’s triangle is 1. Then, recall the connection between consecutive elements in a row in Pascal’s triangle: to move from the th element to the th, we multiply by . Applying this rule, we can calculate the 1st element by multiplying the 0th element (equal to 1) by . Then, to find the second element, we multiply by . Continuing this way, we can find the first five terms in the row, as demonstrated in the figure below.
Notice that once we get to the middle term, we can simply appeal to the symmetry of Pascal’s triangle and fill in the other entries.
Setting and , we have
Finally, we can simplify the numerical terms as follows:
You may be aware that Pascal’s triangle is closely related to the field of combinatorics. Although outside the scope of this lesson, combinatorics provides a more powerful set of tools for understanding and manipulating binomial expansions. You are encouraged to investigate the binomial theorem further, however, the skills in this lesson should provide you with a solid foundation.
- We can quickly reproduce Pascal’s triangle for small to easily expand binomials of the form .
- For a binomial expansion of the form , individual terms can be found by considering coefficients from Pascals triangle in conjunction with successive power of and : where is the row number of Pascal’s triangle and takes integer values from 0 to .
- For larger , we can use the relationship between consecutive terms to expand binomials: to move from the th element to the th, we multiply by .