# Pascal’s triangle – Definition, Patterns, and Applications

Pascal’s Triangle is a number pattern that is known for its shape – yes, a triangle! This interesting pattern and property is named after Blaise Pascal and has been a famous triangle in mathematics due to its extensive application in algebra, number theory, and statistics. Pascal’s Triangle is a number pattern that returns the values or coefficients used in binomial expansions. The numbers in the next layer will depend on the sum of two terms positioned above them in the previous layer. You may already have encountered this unique triangle in your earlier math classes, so in this article, we’ll dive deeper into understanding how we can apply Pascal’s Triangle in algebra and statistics.
For now, let’s refresh our knowledge and take a look at the important properties of a Pascal’s Triangle. What is Pascal’s Triangle?                 The Pascal’s Triangle is a number pattern that shows a significant number of patterns used to expand binomial expressions. Before we dive right into its application in statistics, let’s observe the image shown below. The best way to remember the definition of Pascal’s Triangle and its properties is by constructing the Pascal’s Triangle itself, as shown above. As can be seen, the values in the middle of the third row can be determined by adding the two consecutive terms in the second row. Similarly, the center of the fourth row’s values are the results of the consecutive terms from the third row are added. Here’s an extended version of Pascal’s Triangle shown, and it shows how the sixth row can be formed by: • Using $1$ at the ends of the row. • Adding two consecutive terms to find the term found below them in the next row. The arrows guide the two numbers that were added to find the next row’s term. Pascal’s Triangle definition and hidden patterns Generalizing this observation, Pascal’s Triangle is simply a group of numbers that are arranged where each row of values represents the coefficients of a binomial expansion, $(a+ b)^n$. The rows’ values can be determined by adding two consecutive numbers above each value, as shown in the earlier section. Why don’t we go ahead and arrange Pascal’s Triangle’s values and justify them to the left? This will help us observe the most interesting patterns there are within these numbers. \begin{aligned}&1\\&1\phantom{xxxx}1\\&1\phantom{xxxx}2\phantom{xxxx}1\\&1\phantom{xxxx}3\phantom{xxxx}3\phantom{xxxx}1\\&1\phantom{xxxx}4\phantom{xxxx}6\phantom{xxxx}4\phantom{xxxx}1\\&.\\&.\\&.\end{aligned} We can begin by adding all the terms in each row and observe the result. What can you observe? \begin{aligned}&1\phantom{x} = \phantom{x} 1\\&1\phantom{x}+\phantom{x}1\phantom{x} = \phantom{x}2\\&1\phantom{x}+\phantom{x}2\phantom{x}+\phantom{x}1\phantom{x} = \phantom{x} 4\\&1\phantom{x}+\phantom{x} 3\phantom{x}+\phantom{x}3\phantom{x}+\phantom{x} 1\phantom{x}=\phantom{x} 8\\&1\phantom{x}+\phantom{x} 4\phantom{x}+\phantom{x} 6\phantom{x}+\phantom{x} 4\phantom{x}+\phantom{x}1\phantom{x}=\phantom{x}16\\&.\\&.\\&.\end{aligned} We can see that each row’s sum is actually equal to the power of $2$ that depends on the number of rows.
 Row Number Sum $1$ $2^0 = 1$ $2$ $2^1 = 2$ $3$ $2^2 = 4$ $4$ $2^3 = 8$ $5$ $2^4 = 16$