AUTOMATION OF BINOMIAL EXPANSION USING PASCAL TRIANGLE
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AUTOMATION OF BINOMIAL EXPANSION USING PASCAL TRIANGLE
1.1INTRODUCTION:
Computers are becoming increasingly popular in a variety of applications, and growth is expected to continue at such a rapid pace that only a few institutions will be affected by Binomial Expansion computations utilising the Pascal triangle over the next decade.
The professional position of computing in developing countries examines many gadgets that use computer terms in their native language to impress on radio and television, as well as in films and literature, with or without science fiction.
Many everyday activities in today’s society are performed by computers; some people may be ecstatic at the idea of Automating Binomial Expansion with the Pascal Triangle.
Finally, the computer, as the most recent breakthrough, has raised the standard in every sector of human endeavour.
The Pascal triangle was named after the seventeenth country by mathematician Blasé.
Several subsequent mathematicians were aware of this and had applied their understanding of triangles hundreds of years before Pascal’s birth in 1623. Binomial Expansion in general occurs when a Binomial, such as X+Y, is raised to a positive integer power.
Blaise Pascal’s renditions of the triangle are a collection of numbers that existed before to Pascal’s.
However, Pascal became a frequent user of it and was the first to organise this material together in his work The Triangle Arithmetic in 1653. The number originated with Hindu studies of combinatory and binomial numbers, as well as Greek studies of figurate numbers.
The earliest explicit portrayal of a triangle of binomial coefficient occur in the 10th century in commentaries on the Charles Shactra, and an Indian text on Shanslarit prosody written by Pingalas work only in or before the second century.
Prigal’s work only survives in fragments; the commentator Halanetle, around 1975, uses the triangle to explain observe references to Moruprastara, the staircase of Mount;
many it was also realised that the shallow diagonals of the triangle sum to the Fibonacci number in 1068; and four columns of the first sixteen rows were given by the mathematician Bhatfotpala, who realised the combinatorial significance.
At the same time, it was discovered in Persia (Iran) by mathematician Al-Karaji (953-2029), and it was later repeated by poetastronomer mathematician Omar Khayyan (1048-1131)
so the triangle is known as the Khayyan-pascal triangle or simply the Khayyan. Khayyan used a method of finding roots based on the binomial expansion and thus binomial coefficients.
In Handy, tartaglia is credited with the generic formula for solving cabic polynomials (which may have originated with Delferro but was published by Gerolamo Cordon in 1545).
Pascal’s Trait du Triangle was published posthumously in 1661. In this Pascal collected several results then known about the triangle and employed them to solve problems in probability theorem.
The triangle was later named after Pascal by prime roumond to moutuort (1708) who called it “table de Pascal pour les combination” (French table or Mr. Pascal for combination) and Abraham de move (1780) who called it triangle Arithmetical pascalianum (Latin Pascal’s arithmetic triangle), which became the modern wax term name Binomial Expansion
The primary focus of this research will be on computer-based applications in binomial (Pascal Triangle) that can be automated. The Pascal triangle is a number placed in staggered rows, like this:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 1 1
As it is commonly known that (a + b) 2=a2+2ab+b2, many students find it difficult to write down a similar statement for (a + b) 3 without completing some paper work; nonetheless, it has been recognised that such a solution has significant flaws that may limit its effectiveness.
One of the basic problems most students face is a lack of understanding of how to calculate and write down the binomial coefficient (Pascal Triangle) and enter the number into the computer to display the result. The goal of this research is to learn how the power of (a+b) can be expanded using the computer.
1.2 Statement of the Study
Despite the fact that Binomial Expansion, employing the Pascal triangle, has been used for a long time, individuals still struggle to understand the power and expansion of two numbers (a+b)6 in practically every field.
Sometimes students struggle to write down the Expansion of (a+b) 3 and (a+b) 4 without completing any work on paper. However, it has been recognised that such a method has some drawbacks that impair its efficiency. Lack of knowledge about how to calculate and express the binomial expansion.
1.3 PURPOSES OF THE STUDY
The goal of the research is to create an automated method for solving binomial expansions using the Pascal triangle. The following are some of the specific goals of this initiative.
To create software capable of handling the activity of identifying or solving problems involving binomial expansion.
In contrast to the manual method, this method maintains or improves process accuracy.
To investigate the problems associated with the existing manual system and replace it with an automated system, as well as to improve students’ ability to use computers to solve mathematical problems.
1.4 Significance of the Study
The project was designed to teach students how to use a computer to solve binomial expansion problems using the Pascal triangle.
Binomial Expansion is an important topic for students in various fields of study, particularly science and engineering. Students will profit from this project by reading through the project or text books on how to address the problem topic, and by the end of the project, they will be able to solve binomial expansion using Pascal Triangle.
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