Pascal’s Triangle and its applications and properties

Jordan Leong 3O3 10

History

• It is named after a French Mathematician Blaise Pascal

• However, he did not invent it as it was already discovered by the Chinese in the 13th century and the Indians also discovered some of it much earlier.

• There were many variations but they contained the same idea

History

• The Chinese’s version of the Pascal’s triangle was found in Chu Shi-Chieh's book "Ssu Yuan Yü Chien" (Precious Mirror of the Four Elements), written in AD 1303 which is more than 700 years ago and also more than 300 years before Pascal discovered it. The book also mentioned that the triangle was known about more than two centuries before that.

History

• This is how the Chinese’s “Pascal’s triangle” looks like

What is Pascal’s Triangle

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

Pascal’s Triangle

Simply put, the Pascal’s Triangle is made up of the powers of 11, starting 11 to the power of 0 as can be seen from the previous slide

Interesting Properties

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

In this case, 3 is the sum of the two numbers above it, namely 1 and 2

6 is the sum of 5 and 1

Interesting Properties

• If a line is drawn vertically down through the middle of the Pascal’s Triangle, it is a mirror image, excluding the center line.

Interesting Properties

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

12

5

13

When diagonals Across the triangle are drawn out the following sums areobtained. They follow the formula of X=(3n-1) with n being the number before X

Interesting Properties1

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

1 8 28 56 70 56 28 8 1

In this case, when the triangle is left-justified, the sum of the same coloured diagonals lined out form the Fibonacci sequence

Interesting Properties

• If all the even numbers are coloured white and all the odd numbers are coloured black, a pattern similar to the Sierpinski gasket would appear.

Interesting Properties

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

In this diagonal, counting numberscan be observed

Interesting Properties

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

The next diagonal forms the sequence of triangular numbers.Triangular numbers is a sequence generated from a pattern of dots which form a triangle

Interesting Properties

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

This diagonal contains tetrahedral numbers.It is made up of numbers that form the number of dots in a tetrahedralaccording to layers

Application – Binomial Expansion

• (a+b)2 = 1a2 + 2ab + 1b2

• The observed pattern is that the coefficient of the expanded values follow the Pascal’s triangle according to the power. In this case, the coefficient of the expanded follow that of 112 (121)

Application - Probability

• Pascal's Triangle can show you how many ways heads and tails can combine. This can then show you the probability of any combination.

• In the following slide, H represents Heads and T represents Tails

Application - Probability

• For example, if a coin is tossed 4 times, the possibilities of combinations are

• HHHH• HHHT, HHTH, HTHH, THHH• HHTT, HTHT, HTTH, THHT, THTH, TTHH• HTTT, THTT, TTHT, TTTH• TTTT• Thus, the observed pattern is 1, 4, 6, 4 1

Application - Probability

• If one is looking for the total number of possibilities, he just has to add the numbers together.

Application - Combination

• Pascal’s triangle can also be used to find combinations:

• If there are 5 marbles in a bag, 1 red, 1blue, 1 green, 1 yellow and 1 black. How many different combinations can I make if I take out 2 marbles

• The answer can be found in the 2nd place of row 5, which is 10. This is taking note that the rows start with row 0 and the position in each row also starts with 0.

Purpose

• I chose this topic because while we were choosing a topic for Project’s Day Competition, I researched up on Pascal’s triangle and found that it has many interesting properties. It is not just a sequence and has many applications and can be said to be mathematical tool. Therefore, I decided to explore this now and learned many interesting new facts and uses of the Pascal’s triangle.

Sources

• http://en.wikipedia.org/wiki/Pascal's_triangle• Zeuscat.com• http://www.mathsisfun.com/algebra/triangula

r-numbers.html• http://www.mathsisfun.com/pascals-triangle.

html• http://bjornsmaths.blogspot.sg/2005/11/pasc

als-triangle-in-chinese.html

Thank You