# Mathematics: Pascal's Triangle Essay

# Mathematics: Pascal's Triangle Essay

**Length:** 957 words (2.7 double-spaced pages)

**Rating:** Better Essays

#### Essay Preview

1. (x + y)^0 = 1………………………………………………………………………………. one term

2. (x + y)^1 = x + y………………………………………………………………………… two terms

3. (x + y)^2 = x^2 + 2xy + y2……………………………………………………………. .three terms

Additionally, the integers represented on the triangle are found as the coefficients of the expansion. For instance, the third row of Pascal’s Triangle (0, 1, 2) is “1 2 1”, which corresponds with the coefficients of 3 above. The triangle itself is simply a visualization of this pattern.

The discovery of Pascal’s Triangle is widely considered to have taken place centuries prior to Pascal’s lifetime. However, Blaise Pascal was the first to publish the triangle, which he did in his 1654 work, “Traite du Triangle Arithmetique”. (Kazimir, 2014) Originally, interest in the concept was confined to gamblers as it provided a convenient method to calculate probabilities of an event such as a coin flip or game of dice. (Witchita, 2014) Since then, interest has tended toward the methods and applications of the triangle. The methods of use are essentially number theoretical and the applications are wide. Many fields such as algebra, probability, and combinatorics may find use in Pascal’s Triangle, and additional applications include identifying number sequences such as triangular and tetrahedral numbers.

Each application of Pascal’s Triangle can be solved using all methods available. The convenience of the triangle then is not necessarily in any particular method of examination, but in the variety of method...

... middle of paper ...

...head: head, tail; or tail, head. This yields a 2/4 or 50% chance. Using Pascal’s Triangle for 10 flips, we found the sum of the elements of the 10th row equals 1024. Using combinations, (10 C 5) = 252. Given these results, there are exactly 252 ways to have exactly 5 heads in 10 flips of a coin or (252/1024) = 24.6% chance. Our application of Pascal’s Triangle had given a result, which to some would be a priori counterintuitive.

It is clear to us that Pascal’s Triangle is a very interesting source of useful information. The study of binomial expansions has proven to be a fount of interesting patterns and its allure is the suggestion of more hidden within it. We hope that this cursory examination provides insight for those unfamiliar with Pascal’s Triangle and a renewed interest in those who have experience with it. That is the result which it had on these authors.

## Need Writing Help?

Get feedback on grammar, clarity, concision and logic instantly.

Check your paper »## Essay on Combinations in Pascal's Triangle

- Combinations in Pascal’s Triangle Pascal’s Triangle is a relatively simple picture to create, but the patterns that can be found within it are seemingly endless. Pascal’s Triangle is formed by adding the closest two numbers from the previous row to form the next number in the row directly below, starting with the number 1 at the very tip. This 1 is said to be in the zeroth row. After this you can imagine that the entire triangle is surrounded by 0s. This allows us to say that the next row (row one) is formed by adding 0+1 to equal 1 and 1+0 to equal 1 making the next row 1 1.... [tags: Mathematics Math]

**Better Essays**

894 words (2.6 pages)

## Essay on Who Invented the Pascal Triangle?

- You must have heard of the Pascal triangle, how two numbers above add up to the number below and etcetera etcetera, but do you know the person behind the triangle. Who invented the Pascal triangle. Who turned a calculating machine that only existed in dreams into reality. In this report, we will be investigating, not only about what he invented, but he himself as well. He is Blaise Pascal. Our team had decided to research on a Mathematician, because we believe that there is an inspiring yet neglected story behind every great figure.... [tags: Blaise Pascal, Mathematician, Biography]

**Better Essays**

1912 words (5.5 pages)

## Essay about Blaise Pascal

- Blaise Pascal Blaise Pascal was born on June 19, 1623 in Clermont-Ferrand, France and died August 19, 1662 of stomach cancer. Pascal was not only a great mathematician, he was also a man of great knowledge. Philosopher, mathematician, inventor, and scientific genius are just a few of the things he became in his life. Pascal helped create many things for this world as a whole, like when he invented the barometer, the hydraulic press, and of course (my personal favorite) the syringe. To the mathematical world he was a god, and he is most well known for creating “Pascal’s Triangle” and the adding machine, otherwise known as a calculator (then it was called the “Pascaline.”) As a boy Pasca... [tags: essays research papers fc]

**Better Essays**

490 words (1.4 pages)

## Essay on The Sierpinski Triangle

- The Sierpinski Triangle Deep within the realm of fractal math lies a fascinating triangle filled with unique properties and intriguing patterns. This is the Sierpinski Triangle, a fractal of triangles with an area of zero and an infinitely long perimeter. There are many ways to create this triangle and many areas of study in which it appears. Named after the Polish mathematician, Waclaw Sierpinski, the Sierpinski Triangle has been the topic of much study since Sierpinski first discovered it in the early twentieth century.... [tags: Mathematics Math]

**Better Essays**

1145 words (3.3 pages)

## pascal Essay examples

- Blaise Pascal Blaise Pascal was a French mathematician, physicist, and religious philosopher. He had many important contributions to the mathematics and physics such as: the construction of mechanical calculators, considerations on probability theory, the study of fluids, concepts of the pressure and vacuum, and the Pascal Triangle. After a divine experience in 1654, he devoted himself to meditating and writing philosophy. His many discoveries in the field of mathematics have made him one of the most important mathematicians in history (Broome).... [tags: essays research papers fc]

**Free Essays**

428 words (1.2 pages)

## Essay about Blaise Pascal

- Blaise Pascal was a great mathematician who lived in 17th Century France. His mother was Antoinette Begon; she died when Blaise was three. His father, Etienne took the responsibility of bringing him up. Pascal had two sisters, Gilberte and Jacqueline. Pascal began his studies in 1635 with his reading of Euclid’s Elements and mastered them by age 12. This won the boy respect for his great talent in mathematics. Pascal in fact did not attend school; instead, Etienne brought him to lectures and mathematical gatherings at the “Academie Parsienne” nearly from its founding.... [tags: essays research papers]

**Free Essays**

501 words (1.4 pages)

## Essay about Blaise Pascal

- Blaise Pascal Blaise Pascal was born in Clermont France on June 19, 1623, and died in Paris on Aug. 19, 1662. His father, a local judge at Clermont, and also a man with a scientific reputation, moved the family to Paris in 1631, partly to presue his own scientific studies, partly to carry on the education of his only son, who had already displayed exceptional ability. Blaise was kept at home in order to ensure his not being overworked, and it was directed that his education should be at first confined to the study of languages, and should not include any mathematics.... [tags: essays research papers]

**Free Essays**

600 words (1.7 pages)

## Essay about The French Prodigy

- “Small minds are concerned with the extraordinary, great minds with the ordinary.” Once said by Blaise Pascal. Born in June 19, 1623 in Clermont Ferrand, France, Blaise Pascal was going to change the world of science and mathematics. He was a French mathematician, physicist, inventor, writer, and a Christian philosopher. He was a child prodigy. His early life was no tale of woe or extreme happiness. However, his inventions remain in high esteem. Blaise Pascal made many contributions to the world of math and science.... [tags: mathematics, science, contributions]

**Better Essays**

537 words (1.5 pages)

## Essay on Number Theory

- Research Paper Throughout math, there are many patterns of numbers that have special and distinct properties. There are even numbers, primes, odd numbers, multiples of four, eight, seven, ten, etc. One important and strange pattern of numbers is the set of Fibonacci numbers. This is the sequence of numbers that follow in this pattern: 1, 1, 2, 3, 5, 8, 13, 21, etc. The idea is that each number is the sum of its previous two numbers (n=[n-1]+[n-2]) (Kreith). The Fibonacci numbers appear in various topics of math, such as Pascal?s Triangle and the Golden Ratio/Section.... [tags: Mathematics Math]

**Better Essays**

1699 words (4.9 pages)

## Probability and Genetics Essay

- Probability and Genetics Probability theory is the study of the likelihood of an occurrence of random events in order to predict future behaviors of a system (2). The principles of probability are widely used. In genetics, for example, probability is used to estimate the likelihood of gene distribution from one generation to the next. In business, insurance companies use the principles of probability to determine risk groups. Probability is closely related to statistics since uncertainty always exists when statistical predictions are being made.... [tags: Mathmatics Math Papers]

**Better Essays**

1904 words (5.4 pages)