January 24, 2025

Aashna Chavan

11th Grade

DuPont Manual High School

Pascal’s Triangle is a fascinating mathematical structure composed of numbers arranged in a triangular shape, where each number is the sum of the two directly above it. Starting with a 1 at the triangle’s peak, each subsequent row corresponds to the coefficients in the expansion of binomials. For example, the third row, “1 2 1,” corresponds to the expansion of (a + b)2, expressed as a2  + 2ab + b2. In its elegant construction lie various connections to a plethora of mathematical branches, serving as a fundamental cornerstone of modern mathematics. 

Named after Blaise Pascal, a 17th-century French mathematician, the origins of Pascal’s Triangle predate its namesake by centuries. The earliest known reference to the mathematical structure dates to 11th-century China, where it was already being used in combinatorics as early as the Song Dynasty by Chinese mathematicians. One such mathematician, Jia Xian (1010 AD-1070 AD), is credited with the first recorded use of Pascal’s Triangle, using a triangular array to solve binomial expansion and counting problems. Later, Yang Hui, a 13th-century Chinese mathematician, expanded upon these ideas, with his contribution to the triangle often referred to as the “Yang Hui Triangle.” This mathematical concept quickly spread from China, with mathematicians around the world also utilizing similar concepts to solve mathematical problems. In 12th-century India, mathematicians like Bhaskara II also worked with concepts related to Pascal’s Triangle; Omar Khayyam, a Persian mathematician, explored similar ideas in the Islamic world. However, it was Blaise Pascal who popularized the triangle in 17th-century Europe through his work in combinatorics and probability theory, cementing his association with it. 

Pascal's Triangle reveals a wide array of mathematical patterns and principles that were previously unknown, such as the binomial theorem. Each row in the triangle corresponds to the binomial coefficients in the expansion of (a + b)n, expressing how powers of binomials expand. In addition to binomial expansions, Pascal’s Triangle plays a crucial role in combinatorics, with every number representing a combination. The number in the n-th row and k-th row column ( nk), describes the total amount of ways k objects can be chosen from a set of n objects. With these patterns embedded in the shape’s structure, Pascal’s Triangle serves as an essential tool in counting problems and probability theory. Additionally, the symmetry of Pascal’s Triangle, with the numbers on either side of the center being the same, is not just a visual aspect—it reflects the balanced nature of binomial coefficients and combinational relationships. Certain number sequences within the shape also contain patterns worth noting. The sums of the rows (1, 2, 4, 8, etc.) are powers of 2. The diagonals reveal other interesting sequences, such as triangular numbers (1, 3, 6, 10, etc.) and Fibonacci numbers, or the sum of the diagonals. 

Beyond its number patterns, Pascal’s Triangle has been the inspiration for numerous other mathematical formulas, with the binomial theorem, which expands (a + b)n into a sum of terms involving binomial coefficients, being one of its most important applications. Others include the binomial coefficient formula, which is fundamental to combinatorics and calculates (nk) = n!/k!(n-k)! Additionally, the sum of the numbers in the n-th row of Pascal’s Triangle is equal to 2n, revealing a connection to powers of 2. One well-known identity related to Pascal’s Triangle is the “hockey stick” identity, which states that the sum of numbers along a diagonal of the triangle equals a number further down the triangle.

Pascal’s Triangle is more than just a mathematical curiosity. Its simple construction leads to a wealth of patterns, sequences, and formulas that have had profound implications in algebra, combinatorics, and probability theory. The development of combinatorics introduced the systemic use of binomial coefficients, helping to establish the binomial theorem as a cornerstone of algebra. Pascal’s work on probability theory, including his development of the “probability triangle,” was also influenced by the structure of his namesake triangle. These contributions laid the groundwork for the development of modern probability theory and statistics. Through its elegance and the connections it reveals, Pascal’s Triangle remains an essential part of the mathematical landscape, influencing how we understand and calculate combinations, expressions, and probabilities.