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Pascal’s triangle and its applications and properties

Pascal's Triangle is named after Blaise Pascal, though it was discovered centuries earlier in China and India. It is a triangular array where each number is the sum of the two numbers directly above it. It has many interesting properties, including relationships to binomial expansions, probabilities, combinations, and sequences like Fibonacci numbers. Some of its applications include representing probabilities of coin toss outcomes and finding combinations of objects selected from a group.

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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 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
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 In this case, 3 is the 1 sum of the two numbers 1 1 above it, namely 1 and 2 1 2 1 1 3 3 1 6 is the sum of 5 and 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 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 When diagonals 1 1 2 Across the triangle are drawn out the 1 1 5 following sums are 1 2 1 obtained. They 13 1 3 3 1 follow the formula of X=(3n-1) with n 1 4 6 4 1 being the number 1 5 10 10 5 1 before X 1 6 15 20 15 6 1
Interesting Properties 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 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 1 In this diagonal, 1 1 counting numbers 1 2 1 can be observed 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1
Interesting Properties 1 The next diagonal 1 1 forms the 1 2 1 sequence of triangular numbers. 1 3 3 1 Triangular numbers is 1 4 6 4 1 a sequence 1 5 10 10 5 1 generated from a pattern of dots 1 6 15 20 15 6 1 which form a triangle
Interesting Properties 1 This diagonal contains 1 1 tetrahedral numbers. It is made up of numbers 1 2 1 that form the number of 1 3 3 1 dots in a tetrahedral 1 4 6 4 1 according to layers 1 5 10 10 5 1 1 6 15 20 15 6 1
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/triangul ar-numbers.html • http://www.mathsisfun.com/pascals- triangle.html • http://bjornsmaths.blogspot.sg/2005/11/pasc als-triangle-in-chinese.html
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