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Pascal Triangle

In mathematics, the Pascal’s Triangle is a triangle made up of numbers that never ends. The Pascal’s Triangle was first suggested by the French mathematician Blaise Pascal, in the 17^th century. He had used Pascal’s Triangle in the study of probability theory. After that it has been studied by many scholars throughout the world.

In this section, we will learn what is Pascal’s Triangle, its uses, formula, and properties.

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 Triangle

The topmost row in the Pascal’s Triangle is the 0^th row. The next row below to the 0^th row is 1^st row, and then 2^nd, 3^rd, and so on.

The leftmost element or entry of each row in Pascal’s Triangle is considered as the 0^th element of that row. The elements right to the 0^th elements is the 1^st element of that row, and so on.

From the above image, we have concluded that each i^th row in Pascal’s Triangle contains the i+1 element.

Uses of Pascal’s Triangle

Due 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 Triangle

It 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 Triangle

There 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^th row of the Pascal’s Triangle contains the i+1 element in the row.
• It never ends.

Formula

We can find any entry of the Pascal’s Triangle by using the Combination formula.

In the above formula, n_Cr )means choose r from n. We can also write n_Cr as C(n,r),n^Cr.

Where,

n: is the size of the set (the total number of items in the sample).

C: is the combination.

r: is the subset size (the number of items to be selected from the sample)

Sometimes, we also use k instead of r. Therefore, we can write the above formula in terms of n and k, as:

Let’s find some entries of the Pascal’s Triangle by using the above formula.

Example: Find the entry of 4^th row’s 4^th term.

Solution:

We have to find the entry of the 4^th row’s 4^th term. Here, n=4 and r=4.

Putting the values of n and r in the formula, we get:

Hence, the 4^th term of the 4^th row is 1.

Example: Find the entry of the 6^th row’s 5^th term.

Solution:

We have to find the entry of 6^th row’s 5^th term. Here, n=6 and r=5.

Putting the values of n and r in the formula, we get:

Hence, the 5^th term of the 6^th row is 6.

Similarly, we can find any entry of the Pascal’s Triangle, directly.

Binomial Expansion

In 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 (a+b)ⁿ.

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 (a+b)²=a²+2ab+b²=1a² b⁰+2a¹ b¹+1a⁰ b². In this polynomial, the coefficients are 1, 2, 1.

Similarly, we can also find other binomial expansions.

The following figure shows the first eleven rows of Pascal’s Triangle.

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