Applying Pascal’s Triangle

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Presentation transcript:

Applying Pascal’s Triangle MDM 4U Lesson 5.2

Counting Paths in an Array Ex. #1. Determine how many different paths will spell PASCAL if you start at the top and proceed to the next row by moving diagonally left or right. P A A S S S C C C C A A A L L

Counting Paths Ex. #2: Following the arrows in this directed graph, how many paths are there from A to B? A B

Answer: B A There are 6 paths from A to B. Label each node along the bottom and left sides with a 1 – there is only one way to get to them. For all other nodes, add the numbers from the nodes that lead into them A B 3 1 6 1 3 2 1 1 1 There are 6 paths from A to B.

Ex #3: A diagonal chain- how many paths from A to B.

Ex# 3: Answers B A There are 16 paths from A to B. Is there a pattern? 8 16 4 There are 16 paths from A to B. 4 8 8 3 2 4 Is there a pattern? 4 2 squares paths 1 2 2 4 3 8 4 16 1 2 2 1 1 1 A

Ex# 4: a flat chain- count the paths from A to B.

Ex. #4: Answers B A There are 7 paths from A to B. Is there a pattern? 1 2 3 4 5 6 7 1 2 3 4 5 6 1 A 1 1 1 1 1 1 squares paths 1 2 2 3 3 4 4 5 5 6 Is there a pattern?

Ex. #5: A large graph B A

Ex. #5: Answers B A There are 92 paths from A to B. 92 46 23 46 46 13 18 3 8 23 23 1 3 5 5 5 5 1 There are 92 paths from A to B. 2 2 A 1 1

Counting Paths on a Checkerboard Ex. #6: Count the number of paths on a checkerboard. The checker can travel only diagonally upward. Determine the number of paths from the checker’s current position to the top of the board.

Practice Questions Page 256 #2 – 5, 7, 8