Pascal’S Triangle
Pascal’S Triangle
Binomial numbers or coefficients
Pascal’s triangle, in mathematics, is a geometric arrangement of the binomial coefficients. It is a well-known set of numbers aligned in the shape of a pyramid. The numbers represent the binomial coefficients, which are representations of the number of subsets of a given size. The numbers in Pascal’s triangle are also the coefficients of the expansion of (a + b)n,(a + b) raised to the nth power. So, for n equals to three, the expansion is (a + b) × (a + b)× (a + b), which equals (a2+ 2ab+ b2)× (a+ b), which equals (a3+ 3ab2+ 3ba2+ b3). The coefficients are 1, 3, 3, 1. These are listed in the third row of Pascal’s triangle.
History
Studies of what would be called Pascal’s triangle were done thousands of years before Pascal’s time.
As early as 450 BC, mathematicians in India, Greece, and China were exploring the idea. Later, in Europe, other mathematicians became involved. Pascal’s triangle was also known as the figurate triangle, the combinatorial triangle, and the binomial triangle. The triangle was first given the name Pascal’s triangle by French mathematician Pierre Raymond de Montmort (1678–1719) in 1708. Montmort wrote the numbers in the form below known as the combinatorial triangle.
1 1 1 1 1 1 2 3 4 1 3 6 1 4 1
The combination of numbers that form Pascal’s triangle were well known before Pascal, but he was the first one to organize all the information together in his treatise “The Arithmetical Triangle.” The numbers originally arose from Hindu studies of combinatorics and binomial numbers, and the Greek’s study of figurate numbers. The Chinese also wrote about the binomial numbers in “Precious Mirror of the Four Elements” in 1303. The figurate numbers were known over 500 years BC. There are square and triangular figurate numbers. The first four of each are shown below.
| The Triangular numbers: 1 3 6 10 |
| The Square numbers: 1 4 9 16 |
Additional square and triangular numbers are formed by increasing the size of each, respectively. Actually, figurate numbers can be formed from any polygon. Another set of figurate numbers could be formed using the pentagon, a polygon with five sides. The figurate numbers were studied heavily to learn about counting numbers and arrangements. For example, if a woman was asked to determine which of two sacks of gold coins was worth more, she would probably have to count the coins. To count the coins, the best approach would be to stack the coins into short stacks of a given number. Then, the number of stacks could be counted. Counting numbers, looking at the patterns, and studying the ways objects could be arranged led to the numbers in Pascal’s triangle. The study of combining or arranging objects by various rules to create new arrangements of objects is called combinatorics, an important branch of mathematics. Pascal’s triangle in its current form is shown below. It is the same as the above combinatorial triangle rotated 45 degrees clockwise.
Each new row in Pascal’s triangle is solved by taking the top two numbers and adding them together to get the number below.
The triangle always starts with the number one and has ones on the outside. Another way to calculate the numbers with Pascal’s triangle is to calculate the binomial coefficients, written C(r;c). A formula for the binomial coefficients is r! divided by c!× (r–c)!. The variable r represents the row and c, the column, of Pascal’s triangle. The exclamation point represents the factorial. The factorial of a number is that number times every integer number less than it until the number one is reached. So 4! would be equal to 4× 3× 2× 1, or 24.
Binomial numbers or coefficients
Binomial coefficients are written C(r;c) and represent “the number of combinations of r things taken c at a time.” The numbers in Pascal’s triangle are simply the binomial coefficients. The importance of binomial coefficients comes from a question that arises in every day life. An example is how to take three books from a shelf two at a time. The first two books alone would be one way to take two books from a set of three. The other ways would be to take books two and three or books one and three. This gives three ways to take two books from a set of three. For larger arrangements, listing the number of combinations can be nearly impossible. So instead, the binomial coefficient can be found instead. For the above three books taken two at a time, all that needs to be found in the binomial coefficient C(3,2), which is the third row and second column of Pascal’s triangle, or three.
Pascal
French mathematician Blaise Pascal (1623–1662), a founder of the theory of probability, developed the earliest known calculating machine that could perform the carrying process in addition. The machine, finished in 1642, could add numbers mechanically using interlocking dials. Machines like these eventually led to the first punch card machines and computers. Pascal had a great influence on people like German mathematician Gottfried Wilhelm Leibniz (1646–1716) and English physicist and mathematician Sir Isaac Newton (1642–1727). His father was also a mathematician, and made sure Blaise had the best education possible by introducing him to the Marin Mersenne’s Academy at the age of 14 years. The academy was one of the best places to study mathematics at the time, and his father was one of the founders. When Pascal was young, he was introduced to the work done in combinatorics and the binomial numbers. His paper compiling the work of the Chinese, Hindus, and Greeks would later cause his name to be permanently attached to the combinatorial triangle forever.
Probability theory
A number of unsolved problems in Pascal’s days encouraged the formation of probability theory. The Gambler’s Ruin and the Problem of Point are two examples of such problems.
The Gambler’s Ruin was a problem Pascal challenged French mathematician Pierre de Fermat (1601–1665) to solve. The problem, according to one explanation, was determining what the chances of winning were for each of two men playing a game with two dice. When an 11 was thrown on the dice by the one man, a point would be scored. When the second man threw a 14 on the dice, he would score a point. The points only counted if the opponent’s score was zero. Otherwise, the point scored by one of the men would be subtracted from his opponent’s score. So, one of the men would always have a score of zero throughout the game. The game was won when one man gained 12 points. Pascal asked, what was the probability of each man winning? Binomial coefficients can be used to answer the question.
The Problem of Points was also a game about probabilities. The question was determining how a game’s winnings should be divided if the game was ended prematurely. Questions about games like these stirred the development of probability theory, and the need to understand binomial numbers completely.
Resources
BOOKS
Burton, David M. The History of Mathematics: An Introduction. New York: McGraw-Hill, 2007.
Dickson, Leonard Eugene. History of the Theory of Numbers. Mineola, NY: Dover Publications, 2005.
Edwards, A.W.F. Pascal’s Arithmetical Triangle: The Story of a Mathematical Idea. Baltimore, MD: Johns Hopkins University Press, 2002.
Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.
KEY TERMS
Binomial numbers or coefficients —Numbers that stand for the number of subsets of equal size within a larger set.
Combinatorics —The branch of mathematics concerned with the study of combining objects (arranging) by various rules to create new arrangements of objects.
Pascal, Blaise —Blaise Pascal (1623–1662), a well known mathematician, was a founder of the theory of probability. The combinatorial triangle was given his name when he published a paper compiling the previous work done by the Hindus, Chinese, and Greeks.
Pascal’s triangle —A set of numbers arranged in a triangle. Each number represents a binomial coefficient.
Probability theory —The study of statistics and the chance for a set of outcomes.
Walpole, Ronald, and Raymond Myers, et al. Probability and Statistics for Engineers and Scientists. Englewood Cliffs, NJ: Prentice Hall, 2002.
Waner, Stefan. Finite Mathematics and Applied Calculus. Upper Belmont, CA: Thomson-Brooks/Cole, 2004.
David Gorsich