Pascals Triangle Essay

This essay has a total of 1075 words and 7 pages.

Pascals Triangle



Pascalís Triangle

Blasť Pacal was born in France in 1623. He was a child prodigy and was
fascinated by mathematics. When Pascal was 19 he invented the first calculating
machine that actually worked. Many other people had tried to do the same but did not
succeed. One of the topics that deeply interested him was the likelihood of an event
happening (probability). This interest came to Pascal from a gambler who asked him
to help him make a better guess so he could make an educated guess. In the coarse of
his investigations he produced a triangular pattern that is named after him. The pattern
was known at least three hundred years before Pascal had discover it. The Chinese
were the first to discover it but it was fully developed by Pascal (Ladja , 2).
Pascal's triangle is a triangluar arrangement of rows. Each row except the first
row begins and ends with the number 1 written diagonally. The first row only has one
number which is 1. Beginning with the second row, each number is the sum of the
number written just above it to the right and the left. The numbers are placed midway
between the numbers of the row directly above it.
If you flip 1 coin the possibilities are 1 heads (H) or 1 tails (T). This
combination of 1 and 1 is the firs row of Pascal's Triangle. If you flip the coin twice
you will get a few different results as I will show below (Ladja, 3):


Let's say you have the polynomial x 1, and you want to raise it to some
powers, like 1,2,3,4,5,.... If you make a chart of what you get when you
do these power-raisins, you'll get something like this (Dr. Math, 3):



(x 1)^0 = 1
(x 1)^1 = 1 x
(x 1)^2 = 1 2x x^2
(x 1)^3 = 1 3x 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.
3 * 2 * 1 * 3 * 2 * 1
The triangular numbers and the Fibonacci numbers can be found in
Pascal's triangle. The triangular numbers are easier to find: starting with the third one
on the left side go down to your right and you get 1, 3, 6, 10, etc (Swarthmore, 5)


1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1




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