Pascal's Triangle Essay

Nikko S.A. Gammad

P2C Scicomm

PASCAL’S PYRAMID AND ITS APPLICATIONS IN STATISTICS

Addition, especially of small numbers, is a process that can be done over many repetitions. Sometimes, it produces 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 a pyramid. While Pascal himself did not discover nor popularize it when he was collecting information on the Triangle in the 17th century, the new pattern is still commonly called a Pascal’s Pyramid. Meanwhile, its generalization, like the pyramid, to any number of dimensions n is called a Pascal’s Simplex.

Pascal’s Triangle, because it contains binomial coefficients, is also a representation of the binomial distribution that includes all probability values for finite numbers of experiments with two possible outcomes. Pascal’s Pyramid, then, should represent both the numerical coefficients of the expansion of a trinomial raised to a power and the values in a trinomial expansion.

As it is an extension of another concept, this paper aims first to introduce Pascal’s Triangle and then to apply its properties in deriving each layer of Pascal’s Pyramid. From there, a trinomial principle or theorem is to be formed that will govern the coefficients. For its use in trinomial expansion, the author also aims to compare using any formulas or sho...

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...tries beside each other on the triangle can be added to get the entry below them.

The binomial theorem, according to both Haggard et al and Dhand in his lecture (n.d.), is a way of presenting in an equation the expansion of a power of a binomial (x+y)^n but with all the terms; both numerical and literal coefficients are added. The theorem states:

(x+y)^n=C(n,0) x^n+C(n,1) x^(n-1) y+C(n,2) x^(n-2) y^2+⋯+C(n,n) y^n which, when stated in terms of a summation, becomes

(x+y)^n=∑_(k=0)^n▒〖C(n,k) x^(n-k) y^k 〗.

However, n-k and k can also be represented as a and b to better show the exponents of the two terms of the original binomial in each term of the expansion. The binomial theorem can then be restated:

(x+y)^n=∑_(a+b=n)▒〖n!/a!b! x^a y^b 〗.

Multinomial coefficients are a generalization of the concept for expansions with many terms like (a+b+c+⋯+w+x)^n. In general,