Pascal's triangle

A simple presentation on pascal's triangle covering its patterns and uses.

1. PASCAL’S TRIANGLE MATHS CLUB HOLIDAY PROJECT Arnav Agrawal IX – B Roll.no: 29

2. 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 13 th century and Indians also discovered some of it much earlier. There were many variations but they contained the same idea.

3. Pascal’s Triangle In simple language, the Pascal’s Triangle is made up of the powers of 11 , starting from 11 as shown clearly in the next slide.

4. 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 11 11 1 11 2 11 3 11 4 11 5

5. 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 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

6. 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.

7. 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 If we look at the numbers of the individual rows of this triangle, the sums are doubled for every row. 1 2 4 8 16 32 64

8. Interesting Properties If we draw line segments through each row of the P ascal’s T riangle as shown, and add up the numbers being crossed by it in a row, we will observe the Fibonacci Sequence.

9. 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 appears.

10. 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 In this diagonal, counting number s c an be observed

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

12. 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 This diagonal contains tetrahedral numbers. It is made up of numbers that form the number of dots in a tetrahedral a ccording to layers, also the sums of consecutive triangular numbers.

13. 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

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

15. Sources http://pascalstrianglestuff.blogspot.com http://www.mathsisfun.com/pascals-triangle.html http://ptri1.tripod.com