1. Applying Pascal’s Triangle
MDM 4U Lesson 5.2

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

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

4. 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.

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

6. 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 1 1 1 A

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

8. 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
Is there a pattern?

9. Ex. #5: A large graph B A

10. Ex. #5: Answers B A There are 92 paths from A to B. 92 46 23 46 46 1318 3 8 23 23 1 3 5 5 5 5 1
There are 92 paths from A to B. 2 2 A 1 1

11. 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.

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