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):

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

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