*label*Mathematics

### Question Description

TOPIC: PASCAL TRIANGLE

PLEASE MAKE THE FOLLOWING CHANGES ON THE PASCAL TRIANGLE MATHEMATICS PART ( PROFESSOR SUGGESTIONS) : I see two mathematical ideas related to Pascal's triangle in your paper: how the triangle is constructed and the application of power's of 2. What is your third idea?

PLEASE MAKE THE FOLLOWING CHANGES ON THE REAL WORLD APPLICATIONS PART ( PROFESSOR SUGGESTIONS): You certainly provided a lot of examples of how Pascal’s triangle is applied in the real world. I might encourage you to slim down the list for the final paper submission and talk a bit more about a smaller subset of applications so we can learn a bit more about those applications. You might limit each paragraph to one particular application instead of putting 3 -5 examples in each paragraph. You have a good problem of probably having a little too much information - be selective and tighten up the section a bit. You have the start of something good. Please make sure your final project is in APA formatting.

PLEASE FIX THE CONCLUSION..

ATTACHED IS THE ESSAY RUBRIC.

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Running head: PASCAL TRIANGLE

1

Pascal Triangle

Institution Affiliation

Date

PASCAL TRIANGLE

2

Introduction

The topic selected for this project is “Pascal’s Triangle”. Pascal's triangle is a

mathematical and triangular array of coefficients derived binomially. The triangle is arranged

with the 0th row (n=0) at the top. The entries of the triangle are made from the left-hand side. In

the first row (row 0), the entries are nonzero, which is entry 1 (Lee et al., 2016). The rest of the

entries are made by adding the number above to the left and the right as well and the blank

entries as zero. Pascal's triangle has been used widely and intensively across the world as it's

used spans a wide array.

Pascal's triangle has been used since early days in mathematical contexts especially in

combinatorics and binomial numbers. The triangle has eased the work of mathematicians

especially in the expansion of functions. Pascal's triangle wide usage over years and its

significance has led to setting up of Pascal's rule (Majumdar, 2017). Pascal's triangle with higher

dimensional generalizations has led to the development of Pascal's tetrahedron, while the general

format which is simple is known as Pascal's simplices (Lee et al., 2016). Iranians and Chinese

have had their variants of the same with most of their works being undocumented or put on

record.

Years after it first appeared in Persia and China, the triangle came to be known as

Pascal’s Triangle with Blaise Pascal’s completion of Traité du triangle arithmétique in 1654.

Making use of the already known array of binomial coefficients, French mathematician Pascal

developed many of the triangle’s properties and applications within these writings. Although

Pascal is best known for his work with the arithmetic triangle, he made many other contributions

to mathematics during his lifetime. Throughout his thirty-nine years, Pascal also discovered an

important theorem in geometry, worked with cycloids, invented a calculating machine, laid the

PASCAL TRIANGLE

3

foundations of probability, and planted the seeds of calculus (Eves 242-6). Pascal’s

contributions to mathematics, especially of ‘his’ triangle, were unquestionably brought forth

from the mind of a highly intelligent man.

In mathematical contexts, Pascal's triangle is used in binomial expansions. By using the

triangle, it becomes easy to expand functions up to nth terms. Expanding polynomials is

simplified as it is easy to find the coefficients (Majumdar, 2017). The second mathematical usage

of Pascal's triangle is in binomial probability distributions, which is important to model the

number of successes in a wide sample size that is drawn with instances of replacement.

Mathematics, especially those on polynomials and probability has been improved with the input

of Pascal (Lee et al., 2016). Leibniz's rule of differentiation has also been built and borrowed so

much from the works of Pascal.

The most widely used example of Pascal's triangle in the real-world application is in

gambling and insurance companies. In these contexts, it is used to find out the different

combinations an object may have by using the nCr formulae (Lee et al., 2016). Pascal's triangle

has played a significant role in present-day mathematics with most mathematicians basing and

advancing their work from what Pascal did. In the real world, most mathematical figures have

been derived from the product of the triangle.

Pascal's triangle defined as the set of numbers which precisely arranged in a triangle

containing a certain amount of patterns...

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