PASCAL'S TRIANGLE FOR CLASS XI BENNY VARGHESE P.G.T MATHS J.N.V PUNE Presented By: BENNY VARGHESE P.G.T MATHS J.N.V PUNE ROLL NO: GOA_030_005 23 April 2019 By: GOA_030_005
1 2 + 3 OBJECTIVES 1 4 + 5 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 23 April 2019 By: GOA_030_005
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 23 April 2019 By: GOA_030_005
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 . 23 April 2019 By: GOA_030_005
Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 1 4 6 4 10 5 1 1 5 10 23 April 2019 By: GOA_030_005
How To Form Pascal’s Triangle ? At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. 1 The first row (1 & 1) contains two 1's, 1 1
We may add two cells to get a cell below them 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. 2 3 1 + 2nd ROW
Pascal’s Triangle From A Practical Situation 1
Pascal’s Triangle From A Practical Situation 1 1
Pascal’s Triangle From A Practical Situation
Pascal’s Triangle From A Practical Situation
Pascal’s Triangle From A Practical Situation 1 2 1
Pascal’s Triangle From A Practical Situation
Pascal’s Triangle From A Practical Situation
Pascal’s Triangle From A Practical Situation
Pascal’s Triangle From A Practical Situation
Pascal’s Triangle From A Practical Situation 1 3 3 1
Pascal’s Triangle From A Practical Situation 1 3 3 1
Pascal’s Triangle From A Practical Situation 1 3 3 1
Pascal’s Triangle From A Practical Situation 1 3 3 1
Pascal’s Triangle From A Practical Situation 1 3 3 1
Pascal’s Triangle From A Practical Situation 1 2 1 1 3 3 1
Pascal’s Triangle From A Practical Situation 1 2 1 1 3 3 1
Pascal’s Triangle From A Practical Situation 1 1 1 1 2 1 1 3 3 1
Row Sum Property Of Pascal’s Triangle 1 Sum= 1 = 20 1 1 1 Sum= 2 = 21 1 2 1 Sum= 4 = 22 Sum= 8 = 23 1 3 3 Sum= 16 = 24 1 1 4 6 4 10 5 1 Sum= 32 = 25 1 5 10
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
Magic 11’s & Pascal's Triangle Row 0 Formula Multidigit Number Actual row Row 1 110 1 Row 2 111 11 1 1 Row 3 112 121 1 2 1 Row 4 113 1331 1 3 3 1 Row 5 114 14641 1 4 6 4 1 Row 6 115 161051 1 5 10 10 5 1
Fibonacci Sequence From Pascal’s Triangle
Triangular Numbers From Pascal's Triangle 1 3 6 10
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,…..
I’m extremely grateful to : Hon’ble Dy commissioner, NVS,RO,PUNE Principal,JNV,Canacona Principal,JNV,Pune Mr & Mrs Ekawade, Microsoft