different patterns exhibited in the Pascal’s triangle. One of the main reasons this choice of topic spoke to me is because it relates to a lot of things we do in math class such as Pascal’s triangle, probability, sequences and series, binomial theory, and negative coefficients. Another reason I choose this topic because I am very interested in patterns and I find myself intrigued by patterns and puzzles. It will be interesting to see what I discover. Blaise Pascal was a French philosopher and mathematician
Permutation of Letters EMMA is investigating the amount of different arrangements of letters in her name; she does the same with her friend LUCY. LUCY has twice as many arrangements as EMMA, they are curious as to why this is and decide to investigate other names and find reasons for their answers. EMMA - emma, eamm, emam, aemm, amme, amem, meam, maem, mame, mema, mmea, mmea, LUCY - lucy, luyc, lycu, lyuc, lcyu, lcuy, ulcy, ulyc, uylc,
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
I understand you are taking a college course in mathematics and studying permutations and combinations. Permutations and Combinations date back through the ages. According to Thomas & Pirnot (2014), there is evidence of these mathematical concepts as early as AD 200. As we solve some problems you will see why understanding these concepts is important especially when dealing with large values. I also understand you are having problems understanding their subtle differences, corresponding formulas
Fraction Differences First Sequence To begin with I looked at the first sequence of fractions to discover the formula that explained it. As all the numerators were 1 I looked at the denominators. As these all increased by 1 every time, I figured that the formula was simply [IMAGE] as the denominators corresponded to the implied first line as shown in this table below: nth number 1 2 3 4 5 6 7 8 Denominators 1 2 3 4 5 6 7 8 I shall
represenation a series of binomial expansions. The triangle emerges as a result of the function (x + y) ^n where n is an integer greater than or equal to zero. As n increases, the quantity of terms in the result increases: 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
the fields of mathematics, physical science and computers in countless ways. Blaise Pascal has contributed to the field of mathematics in countless ways imaginable. His focal contribution to mathematics is the Pascal Triangle. Made to show binomial coefficients, it was probably found by mathematicians in Greece and India but they never received the credit. To build the triangle you put a 1 at the top and then continue placing numbers below it in a triangular pattern. Each number is the two numbers
Mathematics make its versatility unparalleled and continues to awe the many Mathematicians today and the many more to come. The Binomial Theorem is one such phenomenon, which was founded by the combined efforts of Blaise Pascal, Isaac Newton and many others. This theorem is mainly algebraic, which contains binomial functions, arithmetic sequences and sigma notation. I chose the Binomial Theorem because of its complexity, yet simplicity. Its efficiency fascinates me and I would like to share this theorem that
interesting patterns. One such pattern is in Pascal’s Triangle, where each row can be constructed by adding the numbers on the row above. This particular pattern is significant in that, among other things, it shows a representation of the coefficients of a binomial expansion to a particular power. There is always room in mathematics, however, for imagination, for expansion of previous concepts. In the case of Pascal’s Triangle, a two-dimensional pattern, it can be extended into a third dimension, forming
Exploring the Binomial Expansion Theorem Introduction In algebra binomial expansion is the expansion of powers of a binomial. A binomial expansion is an expression in which it contains two terms eg, (a+b). This expression could also have a power on the outside of the brackets. Aim To generate a formula for finding the general expanded form of binomial expressions of the form (a+b)n. (Source The Sheet) Basic Binomial Expansions (a+b)1 = a+b (a+b)2 = a2+2ab+ b2 (a+b)3 = a3+ 3a2b + 3ab2 + b3 (a+b)4
+ 3x^2 + x^3 (x+1)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 (x+1)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 ..... If you just look at the coefficients of the polynomials that you get, you'll see Pascal's Triangle! Because of this connection, the entries in Pascal's Triangle are called the binomial coefficients.There's a pretty simple formula for figuring out the binomial coefficients (Dr. Math, 4): n! [n:k] = -------- k! (n-k)! 6 * 5 * 4 * 3 * 2 * 1 For example, [6:3] = ------------------------ = 20.
Blaise Pascal Blaise Pascal was born at Clermont, Auvergne, France on June 19, 1628. He was the son of Étienne Pascal, his father, and Antoinette Bégone, his mother who died when Blaise was only four years old. After her death, his only family was his father and his two sisters, Gilberte, and Jacqueline, both of whom played key roles in Pascal's life. When Blaise was seven he moved from Clermont with his father and sisters to Paris. It was at this time that his father began to school his son
The following paper outlines the use of the Linnaeus system of classification as applied in the field of biology and evolution. The aim of the paper is to highlight how living things are related to other in the ecosystem (Pierce, 2007). It takes us through the evolutionary system highlighting all the important features of life development amongst all the living things. Biological classification Classification is the process of categorizing all the living creatures into group hierarchies citing
Measures of Market Concentration Market concentration describes the extent to which the top firms in an industry, say in the car industry where the top five firms in the UK would account for nearly 90% of the market, take up a large portion of the market share. There are various methods used to measure this, which will be discussed in turn. ‘The concentration ratio is the percentage of all sales contributed by the leading three or five, say, firms in a market.’ (Maunder, P. et al (1991)
Water and all forms of water travel have long fascinated man. With his fascination and the realization that humans are ill-suited for water travel that doesn't involve remaining on the surface, an appreciation for a fish's ability to move in three dimensions with relative ease was also devloped. Although we may not fully understand the physics involved how fish swim, it is obvious from the fascination and the breadth of reseach that it will remain a goal of the modern sicientist. A fish's ability
Gross Domestic Product (GDP) is the market value of all final goods and services produced by factors of production within a country in a given period of time. It can be calculated using either the income, output, or expenditure method as illustrated on the circular flow of income diagram below. Standard of Living, in a purely material dimension is the average amount of GDP per person in a country (therefore determining access to goods and services). However the
VI. Annex i) Global Gini Coefficients from the World Bank: Table 1 Low Income Countries Gini coefficient High Income Countries Gini coefficient Latin American Countries Gini coefficient China 1.6 Australia 1.7 Argentina 2.8 Egypt 1.3 Belgium 1 Bolivia 3.6 India 1.4 Canada 1.4 Brazil 5.6 Ivory Coast 1.6 France 2.1 Chile 4.4 Kenya 4.7 Germany 1.3 Costa Rica 2.5 Madagascar 2.2 Italy 1.4 Ecuador 4.9 Nigeria 2.4 Japan 1 El Salvador 3.5 Pakistan 1.2 New Zealand 1.8 Mexico 4.4 Sri Lanka 1.1 Spain 1 Panama
When I think of Social Welfare, the ideology of Liberalism appears in my head. Not only just the freedom of individuals is important, but the freedom and opportunity to succeed in anything and everything is important. While there is great competition in our economy today, there is also equal opportunity for everyone to thrive and be successful, but nothing can be awarded or given away for free. Personal merits people achieve such as through ones education or promotions in work are the basis of individual
include iron, nickel, copper, zinc and lead. The Behaviour of these metals in terms of partial melting and fractional crystallization is discussed by using their partition coefficients. Partition coefficient is the ratio of concentration of an element in a mixture of a phase relative to another phase. In addition, the partition coefficient depends on pressure, the composition of the solid, the composition of the melt and the temperature (Hall, 1987). Furthermore, the Behaviour of these metals during fractional
model to compare cross-country regressions. Literature Survey Economic Growth and Income Inequality by Simon Kuznets (1955) was one of the initial papers theorizing the long-term effect of inco... ... middle of paper ... ...between the gini coefficient and growth rate of Developing countries is -3.789 with a P-value of .398 and R squared of .132 and .103. Interpreting the statistical significance of the data, the p-values both hold a significance level greater than 0.05, thus we do not reject