Pascal's Triangle is a triangular array of the binomial coefficients formed by taking powers of 11. It was named after French mathematician Blaise Pascal but had been discovered earlier by Chinese and Indian mathematicians. The triangle exhibits several interesting numerical and geometric properties including symmetry, doubling of row sums, generation of the Fibonacci sequence and triangular, tetrahedral and counting numbers along diagonals. It can also be used to calculate probabilities of outcomes from events like coin tosses.

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 13th 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 110 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
110
111
112
113
114
115

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
Pascal’s Triangle 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 numbers
can 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
sequence 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
according 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