Explainer: Pascal’s Triangle

In this explainer, we will learn how to solve problems on Pascal’s triangle.

Pascal’s triangle is one of the most fascinating structures we can build from a simple number pattern. It is fascinating to see the connections between such a simple construction and many other areas of mathematics.

Pascal’s triangle can be formed by starting with a one at the top and then placing two ones below. Then, each element of a row is equal to the sum of the two elements above. Hence, in the figure below, we can see that the two is the sum of the two ones above.

To complete the next row, we can consider the pairwise sum of the elements of this row. The first entry will be 1. We can think of this as the sum of 0 and 1 as shown.

The next element is the sum of 1 and 2 as shown below.

Similarly, the following element is the sum of 2 and 1 as shown.

The final element, like the first, can be thought of as the sum of 1 and 0 as follows.

Continuing this pattern, we arrive at what is known as Pascal’s triangle.

Pascal’s Triangle

Pascal’s triangle is a triangular array of the numbers which satisfy the property that each element is equal to the sum of the two elements above. The rows are enumerated from the top such that the first row is numbered 𝑛=0.

Similarly, the elements of each row are enumerated from 𝑘=0 up to 𝑛. 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 him in places such as China, Persia, and India. To this day, it is know by different names in these places.

Pascal’s triangle has many interesting properties. We will begin by looking as some of the simple patterns which exist in the triangle.

Some of the most obvious patterns are related to the diagonals: for example, the first diagonal only contains ones, whereas the second contains consecutive integers.

More interestingly, the third diagonal contains the triangle numbers, and the fourth contains the tetrahedral numbers.

Furthermore, we can see there is reflectional symmetry about the center.

Example 1: Elements in Pascal’s Triangle

What is the second element in the 500th row of Pascal’s triangle?

Answer

Recall that the second elements of each row of Pascal’s triangle are consecutive integers. At this point, we might be tempted to immediately jump to the conclusion that it will therefore be 500. However, we need to be a little more careful than this. Recall that the first row only contains 1. Hence, there is no second element. The first row with a second element is the second row, which consists of two ones. Therefore, the second element in this row is 1 and not 2. Hence, the second element of the 500th row of Pascal’s triangle will be 499.

Example 2: Patterns in Pascal’s Triangle

A partially filled-in picture of Pascal’s triangle is shown. By noticing the patterns, or otherwise, find the values of 𝑎, 𝑏, 𝑐, and 𝑑.

Answer

We begin by considering the elements of the third diagonal. There is a clear pattern to go from one element to the other: to go from the first to the second, we add two; then to go from the second to the third, we add 3.

We can extend this pattern as follows.

Both 𝑎 and 𝑏 are the elements in this row. Therefore, 𝑎=10 and 𝑏=15.

We now consider element 𝑐. This element is actually also in the third diagonal—the one that is in the other direction—and it is the sixth element. Hence, 𝑐=21.

Finally, we see that 𝑑 is in the second diagonal. This diagonal contains consecutive positive integers. Hence, since it is the eleventh element, its value will simply be 11.

Therefore, our final answer is 𝑎=10, 𝑏=15, 𝑐=21, and 𝑑=11.

Example 3: Sums along Diagonals in Pascal’s Triangle

The figure shows a section of Pascal’s triangle. Without using a calculator, find the sum of the highlighted elements.

Answer

For this question, we could simply sum the individual elements. However, we can actually use the properties of Pascal’s triangle to quickly evaluate the sum of these elements. We will start from the smallest element in the row: the 1. Clearly the sum of this element is simply 1, which we can see is the element below to the right as shown in the figure.

We now consider the first two elements and notice that their sum is the element below the second element to the right.

Similarly, the sum of the first three elements is the sum of the first two elements and the third element. From the defining property of Pascal’s triangle, we see that this is the element directly below these two.