Have you ever heard of the pascal, a unit used to measure pressure? You probably heard of it when your physics teacher bored the entire class with a ridiculously long list of SI units. But who is it named after? As its name might suggest, the pascal is named after French mathematician Blaise Pascal. He made significant contributions to projective geometry at the mere age of 16, and his research into probability theory served as the foundation for modern economics and social science. We know, though, that his triangle is his true claim to fame; the triangle isn’t even his, so to speak: there have been Chinese and Arabian records of such triangles as long ago as the eleventh century. All of this leads to a single question: what is Pascal’s triangle, and what gives the triangle its omnipresence in mathematics?
Despite the ornate term, Pascal’s triangle can be understood with little difficulty. Start by placing 1s on the left and right sides of the triangle. To find the value of a number, simply add the numbers to the left and right of it in the column above. What may be slightly confusing, though, is that Pascal’s triangle starts at the 0th row, similar to how programming languages start counting from 0 instead of 1. For example, row 0 would be a single 1, row 2 would consist of two 1s, row 2 would have 1, 2 (because 1+1), 1, and so on. Repeating the process, you should obtain a triangle similar to the one in the image.
Now that we understand what it is, let’s familiarize ourselves with what the triangle really means. Pascal’s triangle is widely used to find the coefficients of a binomial expansion; to put it into more simple words, each term in the expansion (x+y)n will correspond to a number in the nth row of the triangle. Take (x+y)3 as an example; expanding the expression gives us x3+3x2y+3xy2+y3. This perfectly corresponds to the second row of Pascal’s triangle: 1, 3, 3, and 1. This is also why Pascal’s triangle starts at the 0th row; no matter what x and y is, (x+y)0 will always be equal to 1.
Looking at the triangle itself yields quite a few interesting properties already. Shading in all of the even numbers would result in something like this:
This ever-repeating pattern strongly resembles Sierpinski’s triangle, an equilateral triangle that illustrates the concepts of fractals and recursion. Also, we can align the numbers of Pascal’s triangle to one side and draw diagonals.
Does the sequence look familiar? Based on the fundamental property mentioned earlier, any number in the sequence will be equivalent to the sum of the two previous numbers. Italian mathematician Leonardo Pisano is the first to document and name the sequence, which later became known as Fibonacci’s sequence.
The triangle, though, goes far beyond artistic patterns. The triangle in itself can already be used to derive quite a few interesting properties. For example, we mentioned earlier that each term in the expansion (x+y)n will correspond to a number in the nth row of the triangle. Substituting 1 for both x and y allows us to ignore the variables (I.e., 1(1)2+2(1)(1)+1(1)2 = 1+2+1), which leads us to the result that the sum of the nth row in Pascal’s triangle is equal to 2 to the power of n. We can look back at the triangle to confirm our findings: 1+1 is 2, 1+2+1 is 4, 1+3+3+1 is 8, and so on. However, though, its true potential is yet to be discovered in the realm of combinatorics.
How so? You might ask. How is the triangle related to the selection and arrangement of operations? Let’s look back at Pascal’s triangle. First, denote left with L and right with R; for example, it takes one L to get from the 1 in the third row to the 4 in the fourth row. As you might have noticed, a number in the triangle also indicates the number of ways to reach the number from the top. For example, the second number in the fourth row can be reached by LLRR, RRLL, RLRL, LRLR, LRRL, and RLLR. There are 6 ways, which is to be expected when you look back at the triangle. To make it simpler, though, you can just find the number of ways to arrange 2 Ls and 2 Rs: 4!2!2!. Wait, you might say. Isn’t that the equation for combination? Bullseye. Repeat the process multiple times, and you will find yourself with the simplest proof that the nth number in the mth row in Pascal’s triangle is equal to mCn . Always remember, though, to start counting from 0! That explains why sides are all 1; no matter how many objects you choose from, there is always only 1 way to choose nothing and only 1 way to choose everything.
The property makes Pascal’s triangle very useful in double counting. For example, how can one prove that mCn + mCn+1 is equivalent to m+1Cn+1 for any natural numbers m and n (where m>n)? While other people arduously try to expand the expression, all you had to do is to represent the points using the triangle.
Did you see it? The two sides of the aforementioned equation represent an upside-down triangle similar to the one in the image. And since the bottom value is always the sum of the two values directly above it, the expression is proven to be true - all without writing a single number. Such use of the triangle had created and justified countless theorems, many of which are an absolute nightmare to prove with only algebra.
Okay, quite a handful of people might say. But how does Pascal’s triangle reach beyond math? There are quite a few uses of it other than that involving combinatorics. The binomial coefficients, which are represented by Pascal’s triangle, underlie the fundamental concepts of many programming algorithms and calculus-based approximations. It also has its fair share of applications in real-life situations. If there are three people (named Adam, Bob, and Charlie for convenience) on a basketball court, how many ways are there for Adam to pass the ball and receive it after 6 more passes? This is where Pascal’s triangle comes into play. We represent Adam with red, Bob with ,blue and Charlie with yellow.
The top row represents the initial situation, as the ball starts in Adam’s possession. The first row represents the two possibilities when Adam starts passing: He can either pass to Bob or Charlie. In the next row, the red square is labeled 2 (indicating that there are two possibilities) because both Bob and Charlie can pass to Adam. Continuing the pattern, the seventh row would be coloured as below:
Adding up the numbers, there are 1+20+1 = 22 ways in total for the ball to end up in Adam’s hands after 6 passes. No equations, no tree diagrams, nothing.
The patterns hidden within Pascal’s triangle serve as a testament to the elegance of such combinatorial patterns. To this day, mathematicians are still discovering new uses for the triangle; for one, they discovered a way to expand Pascal’s triangle to multi-variable polynomials (i.e. (x+y+z+...)n). Strive to dive deeper into the intricacy of Pascal’s triangle, and you yourself might be the one to reveal the secrets buried inside of it.
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