Pascal's Triangle Definition
The beauty of Pascal’s Triangle is that it's lucid, yet it is mathematically extremely rich. It is a never-ending equilateral triangular array of numbers. The triangle is formed with the help of a simple rule of adding the two numbers above to get the numbers below it. The two sides of the triangle run down with “all 1’s” and there is no bottom side of the triangles as it is infinite. The first row of Pascal's triangle starts with 1 and the entry of each row is constructed by adding the number above. For example, numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. There are many hidden patterns in Pascal's triangle as described by a mathematician student of the University of Newcastle, Michael Rose. It is surprising that even though the pattern of the Pascal’s triangle is so simple, its connection spreads throughout many areas of mathematics, such as algebra, probability, number theory, combinatorics (the mathematics of countable configurations) and fractals.
Pascal Triangle History
Pascal’s triangle is named after a 17th-century French mathematician, Blaise Pascal, who used the triangle in his studies in probability theory. However, the study of Pascal’s triangle has not only been a part of France but much of the Western world such as India, China, Germany. It also had its presence during the Golden Age of Islam and The Renaissance, which began in Italy before spreading to the rest of the Europe. Jia Xian, a Chinese mathematician in the 11th century devised a triangular representation for the coefficients in the expansion of a binomial expression, such as (x + y)n. Another Chinese mathematician, Yang Hui in the 13th century, further studied and popularized Pascal's triangle. And for this reason, China is often called the Yanghui triangle. It was also included as an illustration in Chinese mathematician Zhu Shijie’s Siyuan yujian, where it was already called the “Old Method.” Pascal’s triangle has also been studied by a Persian poet and astronomer Omar Khayyam during the 11th century.
Construction of Pascal’s Triangle
Step 1: At the top of Pascal’s triangle i.e., row ‘0’, the number will be ‘1’.
Step 2: Keeping in mind that all the numbers outside the Triangle are 0's, the ‘1’ in the zeroth row will be added from both the side i.e., from the left as well as from the right (0+1=1; 1+0=1)
to get the two 1’s in row 1.
Step 3: The 2nd row will be constructed in the same way (0+1=1; 1+1=2; 1+0=1)
The same method will be repeated for every row. There is no bottom
line as the rows of the triangle keep on going infinitely.
Pascal’s Triangle Patterns
Primes: In Pascal’s triangle, you can find the first number of a row as a prime number. This prime number is a divisor of every number present in the row.
Power of 2: Another striking feature of Pascal’s triangle is that the sum of the numbers in a row is equal to 2n.
Magic 11’s: Every row in Pascal’s triangle represents the numbers in the power of 11.
Hockey Stick Pattern: We can even make a hockey stick pattern in Pascal’s triangle. For this, we need to start with any number and then proceed down diagonally. In the end, change the direction of the diagonal for the last number. The last number will be the sum of every other number in the diagonal.
Triangular numbers: If you start with 1 of row 2 diagonally, you will notice the triangular number.
Fibonacci numbers: On taking the sums of the shallow diagonal, Fibonacci numbers can be achieved. A Fibonacci number is a series of numbers in which each number is the sum of two preceding numbers.
Binomial expansion: the coefficients can be found in Pascal’s triangle while expanding a binomial equation. For example, let's consider expanding (x+y)8. To see if the digits are the coefficient of your answer, you’ll have to look at the 8th row. This is true for (x+y)n.
Fractal: You can get a fractal if you shade all the even numbers. Not to be forgotten, this, if you see, is also recursive of Sierpinski’s triangle.
Properties of Pascal Triangle
Listed below are few of the properties of pascal triangle:
Every number in Pascal's triangle is the sum of the two numbers diagonally above it.
All the numbers outside the triangle are ‘0’.
Every row of Pascal's triangle is symmetric.
The first diagonal contains counting numbers.
The sum of the rows of Pascal’s triangle is a power of 2.
In any row of Pascal’s triangle, the sum of the 1st, 3rd and 5th number is equal to the sum of the 2nd, 4th and 6th number (sum of odd rows = sum of even rows)
Every row of the triangle gives the digits of the powers of 11.
Q1: What is the Application of the Pascal Triangle?
Solution: Pascal triangle is used in algebra for binomial expansion. It is also used in probability to see in how many ways heads and tails can combine. Apart from that, it can also be used to find combinations.
Q2: How can we use Pascal's Triangle in Real-Life Situations?
Solution: Pascal's triangle makes the selection process easier. For example, if you have 5 unique objects but you can only select 2, the application of the pascal triangle comes into play. Pascal triangle will provide you unique ways to select them. For another real-life example, suppose you have to make timetables for 300 students without letting the class clash. If there are 8 modules to choose from and each student picks up 4 modules. How many unique combinations will be there? The answer will be 70. Therefore, you need not find a timetable for each of 300 students but a timetable that will work for each of the 70 possible combinations.