Pascal's Triangle

Pascal’s triangle is a fascinating numerical pattern arranged in the shape of a triangle. Pascal’s triangle begins with 1 at the top, and each subsequent row is formed by adding the two numbers directly above it. Pascal’s triangle is a symmetrical triangular structure that contains many interesting mathematical properties and patterns. In this article, we will explore the history, definition, properties, formulas, and examples of Pascal’s Triangle clearly and comprehensively.

Table of Contents

• What is Pascal's Triangle?
• History of Pascal's Triangle
• How to Build Pascal's Triangle
• Pascal's Triangle Formula
• Patterns Inside Pascal's Triangle
• Pascal's Triangle and Binomial Theorem
• Pascal's Triangle in Probability

What is Pascal's Triangle?

Pascal's triangle is a triangle made of numbers, where each number is the sum of the two directly above it. Pascal's triangle is a triangular array constructed by summing adjacent elements in preceding rows, and it contains the values of the binomial coefficients. It is named after the 17th-century French mathematician Blaise Pascal.

 

History of Pascal's Triangle

It's historically important to mention that Pascal's triangle wasn't Pascal's invention. It was his synthesis.

• Indian mathematician Pingala referenced binomial coefficients as early as the 2nd century BCE.

• Chinese mathematician Yang Hui described the triangle in 1261 CE. It is called "Yang Hui's Triangle" in China.

• Persian mathematician Omar Khayyam studied it in the 11th century.

• Blaise Pascal, in 1653, published his comprehensive Traité du triangle arithmétique, in which he thoroughly explained the properties of the triangle. Because of this significant contribution, the triangle came to be known by his name in the Western world.
  
  

How to Build Pascal's Triangle

To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. Each number is the sum of the two numbers directly above it. 

 

Let's look at it row by row:

• Row 0: Just a single 1

• Row 1: Two 1s beneath it → 1 1

• Row 2: The 1s on the edges stay 1; the middle is 1+1 = 2 → 1 2 1

• Row 3: Edges are 1, then 1+2=3 and 2+1=3 → 1 3 3 1

• Row 4: → 1 4 6 4 1

• Row 5: → 1 5 10 10 5 1

The edges are always 1, since there's nothing to the left or right to add, and every other number comes from adding two numbers above it.

The topmost row is called the 0th row. The next row down is the 1st row, and so on. The leftmost element in each row is considered the 0th element.

Pascal's Triangle Formula

The value at any position in Pascal's triangle is given by "n choose k" , written as C(n, k) or  nCk.
  nCk=n!k!(n−k)!

where n is any non-negative integer and 0 ≤ k ≤ n.

"!" means factorial, the product of all whole numbers from that number down to 1

This pattern of getting binomial coefficients is called Pascal’s rule. 

Patterns Inside Pascal's Triangle

1. Symmetry: The triangle is perfectly symmetrical. Every row reads the same from forward and backwards: 1 4 6 4 1, 1 5 10 10 5 1, and so on. 

2. Row sums are powers of 2: The horizontal sums of each row double each time, and they are the powers of 2 because each number in the current row is used twice in the next row. 

 

Row 0: 1 = 20
Row 1: 1+1 = 2 =  21
Row 2: 1+2+1 = 4 = 22
Row 3: 1+3+3+1 = 8 = 23
Row 4: 1+4+6+4+1 = 16 =  24

3. The diagonals: The first diagonal is just 1s. The next diagonal has the counting numbers (1, 2, 3…). The third diagonal has the triangular numbers. The fourth diagonal has the tetrahedral numbers.

Each diagonal is a nesting of sequences within sequences.

4. Powers of 11: Each row is related to a power of 11:  110=1,111=11,112=121,113=1331,114=14641 . Beyond row 4, the digits begin to overlap as they carry over, but the relationship holds. Row 5, for instance, is 1 5 10 10 5 1. Written out with carrying:  115=161,051.  

 

5. The Fibonacci Sequence: The sum of the values along the diagonals is the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21… The diagonals of Pascal's triangle add up to each Fibonacci number in order. 

 

6. Squares of Numbers: For the second diagonal, the square of a number equals the sum of the numbers next to it and below both of those. For example,  32=3+6=9,42=6+10=16,52=10+15=25.

 

7. The Sierpiński Triangle (A Fractal): Coloring all the odd numbers in one color and even numbers in another, you will obtain a pattern similar to the Sierpiński Triangle, a famous fractal shape that repeats itself at every scale. 

 

Pascal's Triangle and Binomial Theorem

Pascal's triangle gives the coefficients in binomial expansion directly.

The expansion of  (x+y)n  is:
  (x+y)n=a0xn+a1xn−1y+a2xn−2y2+…+an−1xyn−1+anyn
where the coefficients of the form ak are precisely the numbers in the nth row of Pascal’s triangle. 

 

Pascal's Triangle in Probability

Pascal's triangle explains how many ways heads and tails can combine. While tossing a coin three times, there is only one way to get three heads (HHH), three ways to get two heads and one tail (HHT, HTH, THH), three ways to get one head and two tails, and one way to get all tails. It is given in the pattern 1, 3, 3, 1 from Pascal's triangle. Let us look into the probability of getting exactly two heads out of four coin tosses. There are 1+4+6+4+1 = 16 possible results, and 6 of them give exactly two heads, so the probability is 6/16, or 37.5%.