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## FOR CLASS XI

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1. PASCAL'S TRIANGLE Presented By: BENNY VARGHESE P.G.T MATHS J.N.V PUNE ROLL NO: GOA_030_005 FOR CLASS XI By: GOA_030_005

2. 1 1 4 2 + + 5 3 OBJECTIVES • To acquire the concept of Pascal’s Triangle and to form it • To have the knowledge of various properties of Pascal’s Triangle • To have the knowledge of application side of Pascal’s Triangle By: GOA_030_005

3. What’s Pascal’s Triangle Pascal’s Triangle is the triangular arrangement of coefficients in the expansion of binomials like (a+b)n for n=0,1,2,3,4,5,6,7,….. Pascal's Triangle is named after Blaise Pascal By: GOA_030_005

4. Pascal’s Triangle In Binomial Expansion (a+b)0 =1 (a+b)1 =1a + 1b (a+b)2 =1a2 +2ab +1b2 (a+b)3 = 1a3 + 3a2b +3ab2 +1b3 (a+b)4 = 1a4+ 4ab3+ 6a2b2+ 4ab3+1b4 and so on . By: GOA_030_005

5. Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 5 10 1 By: GOA_030_005

6. 1 1 1 At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. How To Form Pascal’s Triangle ? The first row (1 & 1) contains two 1's,

7. 1 1 1 1 1 3 1 3 2 1 + + + + 0 0 0 0 0 0 + + + + + + + + We may add two cells to get a cell below them. Thus 2nd row is : 0+1=1; 1+1=2; 1+0=1. In this way, the rows of the triangle go on infinitely. 2nd ROW

8. Pascal’s Triangle From A Practical Situation

9. Pascal’s Triangle From A Practical Situation

10. Pascal’s Triangle From A Practical Situation

11. Pascal’s Triangle From A Practical Situation

12. Pascal’s Triangle From A Practical Situation

13. Pascal’s Triangle From A Practical Situation

14. Pascal’s Triangle From A Practical Situation 1 2 1 1 3 3 1

15. Pascal’s Triangle From A Practical Situation 1 2 1 1 3 3 1

16. Pascal’s Triangle From A Practical Situation 1 1 1 1 2 1 1 3 3 1

17. 1 Row Sum Property Of Pascal’s Triangle Sum= 1 = 20 1 1 1 Sum= 2 = 21 1 2 1 Sum= 4 = 22 Sum= 8 = 23 3 3 1 Sum= 16 = 24 1 4 6 4 1 1 10 5 Sum= 32 = 25 5 10 1

18. Hockey Stick Pattern In Pascal’s Triangle Last number on Hockey stick is the sum of other numbers on it : 1+6+21+56 = 84 1+12 = 13

19. Magic 11’s & Pascal's Triangle

20. Fibonacci Sequence From Pascal’s Triangle

21. QUERY • What do you mean by Pascal‘s Triangle ? • Who discovered the importance of all patterns in Pascal’s Triangle ? • Sum of any row of Pascal’s Triangle is the power of …….. (a) 3 (b) 2 (c) 5 (d) 7 • What’s the name of the sequence 1,1,2,3,5,8,…..

22. I’m extremely grateful to : • Hon’ble Dy commissioner, NVS,RO,PUNE • Principal,JNV,Canacona • Principal,JNV,Pune • Mr & Mrs Ekawade, Microsoft