1. Pascal’s Triangle and its
applications and properties
Jordan Leong 3O3 10

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

3. 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.

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

5. 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

6. 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

7. 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

8. 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.

9. 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

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
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

11. 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.

12. 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

13. 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

14. 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

15. 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)

16. 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

17. 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

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

19. 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.

20. 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.

21. 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

22. Thank You