Pascals Triangle
Pascal’s Triangle Blasé 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 The
Fibonacci numbers are harder to locate. To find them you need to go up at an
angle: you're looking for 1, 1, 1+1, 1+2, 1+3+1, 1+4+3, 1+5+6+1 (Dr. Math, 4).
Another thing I found out is that if you multiply 11 x 11 you will get 121 which
is the 2nd line in Pascal's Triangle. If you multiply 121 x 11 you get 1331
which is the 3rd line in the triangle (Dr. Math, 4). If you then multiply 1331 x
11 you get 14641 which is the 4th line in Pascal's Triangle, but if you then
multiply 14641 x 11 you do not get the 5th line numbers. You get 161051. But
after the 5th line it doesn't work anymore (Dr. Math, 4). Another example of
probability: Say there are four children Annie, Bob, Carlos, and Danny (A, B, C,
D).
