1. PASCAL'S TRIANGLE
AND ITS APPLICATIONS
Adarsh Tiwari
Class- VII-A
Kendrya Vidyalaya Andrews Ganj ,New Delhi-24
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 1

2. Pascal’s Triangle
 Introduction
 Pascal Triangle
 Patterns
 Applications
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 2

3. Blaise Pascal
 French Mathematician born in 1623
 At the age of 19, he invented one of the first
calculating machines which actually
worked. It was called the Pascaline
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 3

4. Pascal's Triangle
 What is a Pascal’s triangle?
 Pascal triangle is algebraic pattern. It was
invented by Blaise Pascal.
 There are many algebraic patterns like
hockey stick pattern, spiral, and Sierpinski
triangle etc. in Pascal's Triangle
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 4

5. Pascal's Triangle
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 5

6. Pascal's Triangle
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 6

7. Fibonacci Series
From Pascal Triangle
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 7

8. Fibonacci Series
 In this series the next term is addition of
previous two numbers.
 the Red line passing through Pascal
Triangle, by addition of the terms of red
line , it results in series called Fibonacci
series . 1,1,2,3,5…….
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 8

9. Golden Ratio /Number
Fibonacci Series is 1,1,2,3,5,8,13
 Golden number is Ratio between two
adjacent terms of Fibonacci series.
 Golden ratio(example 8/5=1.6)
 Example of this ratio we get in natural
Growth like bone growth, plant growth and
building in ancient times.
 It is known as phi / Φ
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 9

10. Spirals
From
Pascal Triangle
 We see spirals around
us in shells, galaxies,
etc.
 This is also drawn
with Fibonacci series.
1,1,2,3,5,8,13……….
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 10

11. Sierpinski Triangle From Pascal Triangle
 From Pascal Triangle we can draw
Sierpinski triangle.
 I have used O for the even numbers and I
for the odd numbers . You can use any
symbol or colors, to get “Sierpinski
Triangle”.
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 11

12. Use of Power in Pascal's Triangle
Power of 2
 First: (2)0 =1
 Second: (2)1 =2
 Third: (2)2 =4
 Forth: (2)3 =8
 Look at the result, they are the
sum of each row of the
Pascal's triangle
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 12

13. Use of Power in Pascal's Triangle
Power of 11
 First: (11)0 =1
 Second: (11)1 =11
 Third: (11)2 =121
 Forth: (11)3 =1331
 Look at the result, they
are the terms combined
together of the Pascal's
triangle
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 13

14. Summing The Rows
1 =1
1 + 1 =2
1 + 2 + 1 =4
1 + 3 + 3 + 1 =8
1 + 4 + 6 + 4 + 1 =16
1 + 5 + 10 + 10 + 5 + 1 =32
1 + 6 + 15 + 20 + 15 + 6 + 1 =64
 
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12  14

15. Binomial Coefficient
 (a+b)*(a+b)=1a*a+2a*b+1b*b
 The numbers which are colored with red
are same as the number in the 3rd row of
the Pascal's Triangle.
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 15

16. Pascal’s Triangle: Row Binomial
coefficients of (1+X)0 (1+X)1 , (1+X)2
 (1+X)0 = 1  1
 (1+X)1 = 1+1X  1 1
 (1+X)2 = 1 + 2X + 1X2  1 2 1
 (1+X)3 =1 + 3X + 3X2 + 1X3  1 3 3 1
 (1+X)4 =1 + 4X + 6X2 + 4X3 + 1X4  1 4 6 4 1
16
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12

17. Hockey Stick Pattern
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 17

18. Hockey Stick Pattern
 The dark numbers
looks like hockey stick.
 To draw Hockey stick
add the numbers of the
longer line , summation
is the left number.
 example- 1+2=3 or
1+1+1+1=4
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 18

19. Symmetry Pascal's Triangle
 You must be familiar with this word
``symmetry”.
 See symmetry in Pascal's triangle.
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 19

20. Symmetry 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
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 20

21. Application
 Pascal triangle is algebraic pattern. From it we
make many pattern like Serpenski Triangle ,
hockey stick pattern ,etc.
 Fibonacci series , 1, 1, 2, 3, 5, 8, can be seen in
the growth in animals plants , shells & spirals.
 Olden Greece buildings used Golden Ratio .
 Binomial coefficients from Pascal Triangle .
 Square numbers 1, 4, 6, 25, 36......
 Counting numbers 1, 2, 3, 4, 5, ......
 Triangular numbers 1, 3, 6, 10, 15........
 Powers of two 1, 2, 4, 8, 6........
 Probability and Games from Pascal Triangle.
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 21

22. Any Questions?
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 22