Pathways and Pascal's Triangle.

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Presentation transcript:

Pathways and Pascal's Triangle

Pascal’s triangle is an array of natural numbers. The sum of any two adjacent numbers is equal to the number directly below them. Sum of each row 1st Row 1 1 20 2nd Row 1 1 2 21 3rd Row 1 2 1 4 22 4th Row 1 3 3 1 8 23 5th Row 1 4 6 4 1 16 24 1 32 25 6th Row 1 5 10 10 5 26 1 6 15 20 15 6 1 64 7th Row 1 7 21 35 35 21 7 1 128 27 8th Row nth Row 2n - 1

Pathways and Pascal’s Triangle Pascal’s triangle can be used to solve pathway problems. A Pascal’s Triangle A 1 1 C 1 2 1 1 B D B 1 1 There is only 1 path from A to C and only 1 path from A to D. There are 2 paths from A to B. 1 2 1 Again, this relates to Pascal’s triangle. 1 3 3 1 This relates to Pascal’s triangle. 1 4 6 4 1 A 1 1 Use Pascal’s triangle to connect the corners of each square for each sum. 2 3 1 3 6 1 B

8C4 = 70 6C2 = 15 Pathways and Pascal’s Triangle Continue with the pattern of Pascal’s triangle to solve larger pathway problems. 1 1 1 1 1 1 A 1 A 1 2 3 4 5 2 3 4 5 1 1 3 6 10 15 3 6 10 15 1 1 B 4 10 20 35 1 5 15 35 70 1 B To simplify these problems, you can use combinatorics: This grid has 4 squares across and 4 squares down. This grid has 4 squares across and 2 squares down. 8C4 = 70 6C2 = 15

Pathways and Pascal’s Triangle Determine the number of pathways from A to B. A A 1. 2. B B 10C5 x 8C3 = 14 112 14C3 x 5C3 x 8C2 = 101 920

Pathways --An Application In a television game show, a network of paths into which a ball falls is used to determine which prize a winner receives. a) How many different paths are there to each lettered slot? b) What is the total number of paths from top to bottom? There is only one pathway to each of Slots A and F. There are five pathways to each of Slots B and E. There are ten pathways to each of Slots C and D. The total number of pathways from top to bottom is 32. (Row 6 of Pascal’s triangle, n = 5: 25 = 32) 1 5 10 10 5 1 The total number of pathways from top to bottom is 128. Determine the number of pathways from top to bottom for this network. (Row 8 of Pascal’s triangle, n = 7: 27 = 128)