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1.11 Pascal's Triangle.notebook September 20, 2013 1.11 Pascal's Triangle Position 1 Position 0 Row 0 Pascalʹs Triangle is built with an iterative process. Each term is equal to the sum of the two terms directly above it. The first and the last term in each row are both equal to 1. Row 1 Row 2 Row 3 Row 4 t3,0 t3,1 t4,1 t0,0 t1,0 t2,0 t3,0 t1,1 t2,1 t3,1 Notation: t2,2 t3,2 etc... t3,3 termrow,position tn,r 1 1.11 Pascal's Triangle.notebook September 20, 2013 Pascal's Identity The following relationship, known as Pascal's Identity, is often useful. tn,r represents the term in row n, position r. Thus, tn,r = tn1,r1 + tn1,r Example 1 Express as a single term from Pascal's Triangle. a. t6,3 + t6,4 b. t14,5 + t14,6 c. t21,7 ‐ t20,6 d. t15,10 ‐ t14,10 Example 2 The 1st six numbers in row 25 are 1, 25, 300, 2300, 12650, 53130. Determine the first six numbers of row 26. 2 1.11 Pascal's Triangle.notebook September 20, 2013 Row Sums The sum of the terms in any row n in Pascal's Triangle is 2 n. eg. sum of row 9 = 29 = 512 eg. row whose sum is 4096 is row 12, ∵ 212 = 4096 Example 3 What is the sum of row 7? Example 4 Which row in Pascal's triangle has a sum of 32,768? 3 1.11 Pascal's Triangle.notebook September 20, 2013 Combinations Is there any link between Pascalʹs triangle and Combinations? Recall: t0,0 n ( ) = nCr = C(n,r) r t1,0 t2,0 t3,0 t1,1 t2,1 t3,1 t2,2 t3,2 t3,3 etc... Example 5 Determine the first 5 numbers of the 18th row of Pascal's triangle. Example 6 The 6th and 7th term of a row is 792 & 924. Which row? 4 1.11 Pascal's Triangle.notebook September 20, 2013 Triangle Numbers Coins can be arranged in the shape of an equilateral triangle. Determine the number of coins in the triangle when there are four, five and six rows. Locate these numbers in Pascal's triangle to the number of coins in a triangle with n rows. Example 7 How many coins are there in a triangle with 12 rows? 5 1.11 Pascal's Triangle.notebook September 20, 2013 11 and Pascal Relate 11 to Pascals' Triangle 6 1.11 Pascal's Triangle.notebook September 20, 2013 Applying Pascal's Method Pascal's Triangle is often used to determine the number of paths between points. Example 8 Determine how many different paths will spell SUCCESS if you start at the top and proceed to the next row moving diagonally left or right. S U U C C C C C C C E E E S S S Example 9 In Plinko, a token slides down a board. If the token cannot go through a shaded square, in how many ways could you win $500? START $500 $500 7 1.11 Pascal's Triangle.notebook September 20, 2013 8