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. The second row is formed using the same rule so 0+1=1, 1+1=2, and 1+0=1 making this row 1 2 1. This rule can continue on into infinity making the triangle infinitely long.
Pascal’s Triangle falls into many areas of mathematics, such as number theory, combinatorics and algebra. Throughout this paper, I will mostly be discussing how combinatorics are related to Pascal’s Triangle.
Yang Hui has been found to be the oldest user of Pascal’s Triangle. But it is Blaise Pascal who around the year 1654 was credited for his extensive work on the many patterns of this triangle. Because of this people began to call it Pascal’s Triangle.
To demonstrate how combinations of elements of a set are showing up in Pascal’s Triangle, I will use the model on page 4. I set up this model by first placing blank boxes
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on the entire left side of the triangle (column 3) to represent n things going into groups of 0. There is only one way to do this, so every cell with a blank box can be said to have one item in it. I am using a box because it does not have a specific “value” but it is more of a holding place for new elements. In cell (B, 4), I placed an “a” because this represe...
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... the new possibility for 3 things going 3 at a time: “ab” combined with “c” becomes “abc.”
Below the chart that we have been following is a form of the original Pascal’s triangle that corresponds to the chart above it. If you count up all the pieces in a particular cell, you will see that their sum is equal to the corresponding cell in the recreated Pascal’s triangle.
In conclusion I would like to say that this discussion was not designed to be a proof of why combinations exist but an explanation of how these patterns occur. As you think about how combinatorics show up in Pascal’s Triangle, keep in mind that this is just one of the many patterns that are concealed within this infinitely long mathematical triangle.
Works Cited
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Pascal’s Triangle. Jessica Kazimir. Montclair State University. 28 July 2005. .
On the second day of class, the Professor Judit Kerekes developed a short chart of the Xmania system and briefly explained how students would experience a number problem. Professor Kerekes invented letters to name the quantities such as “A” for one box, “B” for two boxes. “C” is for three boxes, “D” is for four boxes and “E” is for five boxes. This chart confused me because I wasn’t too familiar with this system. One thing that generated a lot of excitement for me was when she used huge foam blocks shaped as dice. A student threw two blocks across the room and identified the symbol “0”, “A”, “B”, “C”, “D”, and “E.” To everyone’s amazement, we had fun practicing the Xmania system and learned as each table took turns trying to work out problems.
Sum Law (the sum of the interior angles of a triangle must sum to 180
The last activity that we did was taking ten Q tips and made three attached squares and her assignment was to make a 4th enclosed box without adding an additional items. Once I told her to start she immediately started moving the Q tips around trying to create another box. After trying for a few minutes she then say there is no way to add another box.
The 2's and 5's were arranged in such a way that one number formed a distinct shape in the midst of the jumble of the other number. A non-synesthetic would be incapable of distinguishing any pattern due to the close resemblance of the numbers. But, in 90% of the cases where people claimed to see colors they were easily able to discern the shape because it registered stood out for them as a completely different color.
Blaise Pascal was born on 19 June 1623 in Clermont Ferrand. He was a French mathematician, physicists, inventor, writer, and Christian philosopher. He was a child prodigy that was educated by his father. After a horrific accident, Pascal’s father was homebound. He and his sister were taken care of by a group called Jansenists and later converted to Jansenism. Later in 1650, the great philosopher decided to abandon his favorite pursuits of study religion. In one of his Pensees he referred to the abandonment as “contemplate the greatness and the misery of man”.
First we are going to talk about probability theory, which has to do with mathematics and analysis of random phenomena. You are probably used to putting the number of outcomes over the total amount of the object or total amount what you have. An example is, if you have a normal dice and you want the probability of rolling an odd number, you would take the total amount of odd numbers (3) and put that over the total (6) amount of numbers on the dice like so 3/6 which you can also reduce it to ½ because 3 is half of 6. This theory has been around since the sixteenth century and started off as the outcome you would get in a game, which was created by Pierre de Fermat, Blaise Pascal and Gerolamo Cardano. Later on in the seventeenth century Christiaan Huygens published a book on the subject.
In 1886 Dorr D. Felt (1862 - 1930) invented the "comptometer". This was the first calculator where the operands are entered by just pressing keys. In 1889 in also invents the first printing desk calculator.
The proof for the existence of God is an issue that may never be resolved. It has caused division among families and friends, nations and society. The answer to the question “does God exist?” is almost an impossible one to give with certainty seeing that there is a variety of people, ideas, cultures and beliefs. So how does one know if one’s actions here on earth could have eternal consequences? What is, if any, a “safe bet” to make? Blaise Pascal was a 15th century philosopher and a mathematician who proposed the idea that although one cannot know for certain that God exists, one can make a “safe bet” that it is far better to believe in God than not to believe in God. This is not a proof for the existence of God but rather an idea that suggest that if there is a God, it is in the person’s benefit to believe rather to disbelieve because the odds are in favor of the believer. This gambler-like idea is better known as “Pascal’s Wager” or “The Gambler’s Argument.” Nevertheless, this sort of play-the-ponies idea is not quite precise. Although Pascal’s Wager serves as a stepping-stone for non-believers, it is a rather vague, faithless and inaccurate argument.
Blaise Pascal was born on June 19, 1623. Pascal was a mathematician along with a Christian philosopher who wrote the Pensees which included his work called Pascal’s wager. The crucial outline of this wagers was that it cannot be proved or disprove that God does exists. There are four main parts to the wager that include his reasoning to that statement. It has been acknowledged that Pascal makes it clear that he is referring to the Christian God in his wager. This is the Christian God that promises his people will be rewarded with eternal life along with infinite bliss.
... Pascal was such a brilliant man because he could do both of these. Pascal was one of the only men that wrote about his beliefs in God and was an accredited scientist and mathematician too. He was a true man of the scientific revolution.
Neale, Vicky. "Theorum11the pigeonhole principle." Theorem of the Week. Cambridge Maths Tripos, 25 March 2009. Web. 8 Dec 2013. .
Fibonacci was born in approximately 1175 AD with the birth name of Leonardo in Pisa, Italy. During his life he went by many names, but Leonardo was the one constant. Very little is known of his early life, and what is known is only found through his works. Leonardo’s history begins with his father’s reassignment to North Africa, and that is where Fibonacci’s mathematical journey begins. His father, Guilielmo, was an Italian man who worked as a secretary for the Republic of Pisa. When reassigned to Algeria in about 1192, he took his son Leonardo with him. This is where Leonardo first learned of arithmetic, and was interested in the “Hindu-Arabic” numerical style (St. Andrews, Biography). In 1200 Leonardo ended his travels around the Mediterranean and returned to Pisa. Two years later he published his first book. Liber Abaci, meaning “The Book of Calculations”.
The history of the computer dates back all the way to the prehistoric times. The first step towards the development of the computer, the abacus, was developed in Babylonia in 500 B.C. and functioned as a simple counting tool. It was not until thousands of years later that the first calculator was produced. In 1623, the first mechanical calculator was invented by Wilhelm Schikard, the “Calculating Clock,” as it was often referred to as, “performed it’s operations by wheels, which worked similar to a car’s odometer” (Evolution, 1). Still, there had not yet been anything invented that could even be characterized as a computer. Finally, in 1625 the slide rule was created becoming “the first analog computer of the modern ages” (Evolution, 1). One of the biggest breakthroughs came from by Blaise Pascal in 1642, who invented a mechanical calculator whose main function was adding and subtracting numbers. Years later, Gottfried Leibnez improved Pascal’s model by allowing it to also perform such operations as multiplying, dividing, taking the square root.
There is a triangle called the Heronian triangle. It has area and side lengths that are all integers. The Heronian triangle is named after the great hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers. An Isosceles triangle is a triangle that has two sides of equal length. Sometimes is specified as having two and only two sides of equal length. Triangles are polygons with the least possible number of sides, which is
The Fibonacci Series was discovered around 1200 A.D. Leonardo Fibonacci discovered the unusual properties of the numeric series, that’s how it was named. It is not proven that Fibonacci even noticed the connection between the Golden Ratio meaning and Phi.