INTRODUCTION
Blaise Pascal (1623-1662), a French mathematician, discovered a pattern now known as Pascal's Triangle of Coefficients.
In mathematics, Pascal's triangle is triangular array of binomial coefficient.
The coefficients May be written in either way.
How to Build it?
✓ First of all start with "1" and second rows as "1 1" then we can clearly see every element in next rows(except 1) if sum of two elements from above row. (i.e, 1+1=2, 1+2=3, 1+3=4) and with this rules we can create pascal's triangle of n rows.
✓let's take 0th row as first row and 0th column as first column, so we can get each value using nCk formula where n is row number and k is column number. so for finding 1st(0 indexed) element in 2nd row we can write 2C1 which will give 2.
✓There is one another technique, in which we need to solve (x + y)^n for nth row, if we solve (x +y)² then we will get 2nd row as coefficient of x and y in this solved formula which is x² + 2xy + y² . coefficients are (1 ,2 , 1) which is second row in pascal's triangle.
To help you understand better, you can watch the link below which explains how to use the binomial theorem to foil and expand binomial expressions using pascal's triangle and combinations. This video also shows you how to find the nth term and its coefficient using a simple formula / equation.
https://youtu.be/s19dWIHficY?si=V5VWEkM1E-G7MOQW
Some amazing facts
✓if you have noticed if we sum of element in row then we'll get 2^n , where n is row number in that pascal's triangle. ( 1 = 2° , 1+1 = 2 , 1+2+1 = 2² ).
✓if we take whole row as one number then it is power of 11, ( 11° = 1, 11 = 1 1, 11² = 1 2 1).
CONCLUSION
The patter may be continued indefinitely. Observe that each row in the expansion begins and ends with 1. The second number in any row is obtained by adding the first and second numbers in the preceding row While the third number is obtained by adding the second and third numbers in the preceding row, and so forth.
Brief History:
Pascal’s triangle, in algebra, a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y)^n.
Illustration:
It is named for the 17th-century French mathematician Blaise Pascal, but it is far older. Chinese mathematician Jia Xian devised a triangular representation for the coefficients in the 11th century. His triangle was further studied and popularized by Chinese mathematician Yang Hui in the 13th century, for which reason in China it is often called the Yang Hui triangle. It was included as an illustration in Chinese mathematician Zhu Shijie’s Siyuan yujian (1303; “Precious Mirror of Four Elements”), where it was already called the “Old Method.” The remarkable pattern of coefficients was also studied in the 11th century by the Persian mathematician, astronomer, and poet Omar Khayyam.
The triangle can be constructed by first placing a 1 (Chinese “—”) along the left and right edges. Then the triangle can be filled out from the top by adding together the two numbers just above to the left and right of each position in the triangle. Thus, the second row, in Hindu-Arabic numerals, is 1 1, the third row is 1 2 1, the fourth row is 1 3 3 1, the fifth row is 1 4 6 4 1, the sixth row is 1 5 10 10 5 1, and so forth. The first row, or just 1, gives the coefficient for the expansion of (x + y)^0 = 1; the second row, or 1 1, gives the coefficients for (x + y)^1 = x + y; the third row, or 1 2 1, gives the coefficients for (x + y)^2 = x^2 + 2xy + y^2; and so forth.The triangle displays many interesting patterns. For example, drawing parallel “shallow diagonals” and adding the numbers on each line together produces the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21,…,), which was first noted by the medieval Italian mathematician Leonardo Pisano (“Fibonacci”) in his Liber abaci (1202; “Book of the Abacus”).
Reference
Soledad Jose-Dilao, and Julieta G. Bernable. Algebra II, SD Publications, Inc.
https://community.codenewbie.org/user64bit/pascals-triangle-974
https://www.amazon.co.uk/Blaise-Pascal-Legendary-Mathematician-Theologian/dp/1729791352#
https://www.mathsisfun.com/pascals-triangle.html
https://youtu.be/s19dWIHficY?si=V5VWEkM1E-G7MOQW
