## Pascal TriangleIn mathematics, the In this section, we will learn ## What is Pascal's Triangle?Pascal's Triangle is a never-ending equilateral triangle in which the arrays of numbers arranged in a triangular manner. The triangle starts at 1 and continues placing the number below it in a triangular pattern. Remember that Pascal's Triangle never ends. In Pascal's Triangle, each number is the sum of the two numbers above it. ## Notation of Pascal's TriangleThe topmost row in the Pascal's Triangle is the ^{th} row is 1^{st} row, and then 2^{nd}, 3^{rd}, and so on.The From the above image, we have concluded that each i+1 element.## Uses of Pascal's TriangleDue to its simple and easy pattern, it is used in many areas of mathematics such as probability, algebra, number theory, combinatorics, and fractals. It is also used to find the coefficients of polynomials. ## How to Find the Entries of the Pascal's TriangleIt is a triangle that follows the rule of adding two numbers together. To get a new number of the triangle we add the numbers above the determining row (for which we are calculating the numbers). In the following figure, the pointed entries show the sum of two entries above to it. ## Properties of Pascal's TriangleThere are the following properties of the Pascal's Triangle: - It is a
**symmetric**It means the triangle has a mirror image. In the following figure, the red color entries show the symmetry. - The leftmost and the rightmost entry of each row is
**1**. In the following figure, the red and green color entries are left and right-most entries of Pascal's triangle that are 1. - The
**first**diagonal (blue) and the last diagonal shows the**1's**. - The
**second**diagonal (pink) shows the**counting numbers,**such as 1, 2, 3, 4, etc. - The
**third**diagonal (yellow) shows the**triangular numbers**, such as 1, 3, 6, 10, etc. - The sum of the rows gives the power of 2. The power is corresponding to the row number.
- Each row is the power of 11. The power is corresponding to the row number.
- Each row starts and ends with
**1**. - In the second diagonal, the square of each number is equal to the sum of the number next to it and below it, as we have shown in the following figure.
- The
**i**row of the Pascal's Triangle contains the^{th}**i+1**element in the row. - It never ends.
## FormulaWe can find any entry of the Pascal's Triangle by using the In the above formula, n as _{Cr}C(n,r),n.^{C}rWhere,
Sometimes, we also use Let's find some entries of the Pascal's Triangle by using the above formula.
We have to find the entry of the 4 Putting the values of n and r in the formula, we get:
We have to find the entry of 6 Putting the values of n and r in the formula, we get:
Similarly, we can find any entry of the Pascal's Triangle, directly. ## Binomial ExpansionIn algebra, binomial is a term that is used to add two things together. It refers to the pattern of coefficients (the number before the variables). When we multiply the binomial by itself a certain number of times, we get the coefficients. It is written as With the help of Pascal's Triangle, we can also determine the coefficients of a binomial expansion. Consider the second row's polynomial expansion that is 1, 2, 1.Similarly, we can also find other binomial expansions. The following figure shows the first eleven rows of Pascal's Triangle. Next TopicPerimeter of Rectangle |