5 Interesting Facts About the Pascal’s Triangle

by Carson
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Pascal's triangle

Pascal’s triangle is arguably one of the most beautiful patterns in mathematics. It is so simple but has very intriguing specialties that you can observe yourself. Here are 6 interesting facts about Pascal’s triangle, so let’s find out about them.

Table of Contents

  1. How the Pascal’s triangle goes on
  2. Binomial coefficients
  3. Triangular numbers
  4. The powers of 2
  5. The sum of nth powers

1. How the Pascal’s Triangle Goes On

Unlike the list of prime numbers, which is not very well-defined because of the lack of an algorithm to accurately predict the next one, Pascal’s triangle goes on in a surprisingly easy-to-follow way. Imagine an infinitely large space tessellated with regular hexagons and assign one of them with a value of 1, and all other hexagons are assigned a value of zero. Next, point out the two hexagons right beneath the initial hexagon, and assign them values by obtaining the sum of the values of the two hexagons above them. Specifically, they all have values of 1 since one of the hexagons are assigned a value of 1, while the other has a value of 0. If you find this piece of text hard-to-understand, the illustration may help you.

An illustration on the process of how the Pascal’s triangle goes on forever
Image created using Canva

2. Binomial Coefficients

Although Pascal’s triangle may look like it’s nothing special simply because it is easy to construct, that is by no means true. One of the patterns that can be found on Pascal’s triangle is the list of binomial coefficients, which is essentially the number of possible options to choose k items from n items or n choose k. They are defined as the result of a function shown below.

\[n! \over k!(n-k)!\]

3. Triangular Numbers

Another feature that you can see from Pascal’s triangle is that every third integer from the second row is the list of triangular numbers. This is extremely easy to explain because the nth triangular number is the sum of the first n integers. Although the second row may seem like it’s the third row, Pascal’s triangle is actually zero-indexed, which means that the row at the very top is the zeroth row, not the first row.

Did you notice that the list of all integers is in the second item of each row, starting from the first row of Pascal’s triangle? That way, the third integer in the nth row will be the sum of the second integer, which is simply an item in the list of integers, and the third item, which is the previous triangular number, all from the (n-1)th row.

The first 5 rows of Pascal’s triangle. The topmost row is the zeroth row. The integers marked in red correspond the triangular numbers.
Image created using Canva

4. The Powers of 2

The sum of all numbers in the first row of Pascal’s triangle is 1, the sum of all integers in the second row is 2, for the third row, it’s 4, and for the fourth row, it’s 8. Did you notice something interesting in this pattern? The sum of a row of integers in Pascal’s triangle is precisely a power of 2. Specifically, for the nth row of this triangle, the sum of all integers in it is 2 to the power of n-1.

This is also a property that is very simple to prove precisely by how Pascal’s triangle goes on. When the next row of the triangle is computed, all positive integers in the last row are used twice to participate in the way to acquire the two numbers below each integer. Thus, the sum of all integers in a row doubles with each row of Pascal’s triangle.

The first 5 rows of Pascal’s triangle. Integers marked in red are the sum of the rows of this triangle and are powers of 2.
Image created using Canva

5. The Sum of Nth Powers

Imagine that you have to expand this polynomial (shown below).

\[(m + n)^8\]

It’s probably too challenging for the average reader to directly expand this polynomial without making mistakes. However, there is a better way to do so. It involves using the Pascal’s triangle, or more specifically, binomial coefficients. Let’s construct the polynomial for the rest of the section.

Firstly, take the eighth row of Pascal’s triangle. For the sake of brevity, we will simply list all integers in the row, which are 1, 8, 28, 56, 70, 56, 28, 8, and 1. After that, start by 1 multiplied by m8 multiplied by n0. Since any number to the power of zero is equal to 1, we omit the n0 and the 1. Therefore, the first term of this polynomial is m8. Similarly, we can construct the second term by multiplying 8, the number after 1 in the 8th row of Pascal’s triangle, m7, which is essentially m8 divided by m, and n, which is n0 multiplied by n. Continue this process until the last term is reached. If you don’t understand the process, the following result may help you.

\[m^8 * 8m^7n * 28m^6n^2 * 56m^5n^3 * 70m^4n^4 * 56m^3n^5 * 28m^2n^6 * 8mn^7 * n^8\]

Conclusion

In this article, we’ve explained 5 patterns hidden in Pascal’s triangle. This pattern is so easy to compute but is the key to so many valuable properties that may seem entirely unrelated to addition, which is in the core of the triangle.

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