# Mathematics: Pascal's Triangle Essay

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Pascal’s Triangle is a visual 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. 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.

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