Pascal's triangle
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PASCAL’S TRIANGLE MATHS CLUB HOLIDAY PROJECT Arnav Agrawal IX – B Roll.no: 29
Transcript of Pascal's triangle
PASCAL’S TRIANGLE
MATHS CLUB HOLIDAY PROJECT Arnav AgrawalIX – BRoll.no: 29
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 Indians also discovered some of it much earlier.
• There were many variations but they contained the same idea.
PASCAL’S TRIANGLE
In simple language, the Pascal’s Triangle is made up of the powers of 11, starting from 110 as shown clearly in the next slide.
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
110
111
112
113
114
115
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
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
INTERESTING PROPERTIES
If we draw line segments through each row of the Pascal’s Triangle as shown, and add up the numbers being crossed by it in a row, we will observe 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 appears.
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 numberscan be observed
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 sequence of triangular numbers.Triangular numbers is a sequence generated from a pattern of dots which form a triangle
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 tetrahedralaccording to layers, also the sums of consecutive triangular numbers.
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
• If one is looking for the total number of possibilities, he just has to add the numbers together.
SOURCES
• http://pascalstrianglestuff.blogspot.com
• http://www.mathsisfun.com/pascals-triangle.html
• http://ptri1.tripod.com
