1. 4.2 Pascal’s Triangle

2. Consider the binomial expansions… Let’s look at the coefficients… 1 11 121 1331 14641 1 510 51

3. Pascal’s Triangle 1 11 121 1331 14641 1510 5 1 Etc.

4. Pascal’s Triangle and Paths… 1 11 121 1331 14641 1510 51 How many paths can you take to get to the indicated point? 2 2 ways

5. How many paths can you take to get to the indicated point? Pascal’s Triangle and Paths… 1 11 121 1331 14641 1510 51 4 4 ways

6. Pascal’s Triangle and Paths… 1 11 121 1331 14641 1510 51 How many paths can you take to get to the indicated point? 10 10 ways 1 1 way 10

7. Example 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 S S S C C A A A L Write the triangle coefficients 1111 2 11 3 3 464 10 There are 10+10 = 20 paths that will spell PASCAL.

8. Example 2 On the checker board shown, the checker can travel only diagonally upward. It cannot move through a square containing an X. Determine the number of paths from the checker’s current position to the top of the board. X There are 55 paths the checker can take to get to the top.

9. Example 3 How many paths are there from school to Harvey’s (assume that you don’t double-back)? School Harvey’s There’s another way to do this… There are 35 ways.

10. Example 3 Note: there are 7 blocks in total to travel: 4 going east, School Harvey’s 3 north.

11. Example 3 Note: there are 7 blocks in total to travel: 4 going east, 3 north. OR School Harvey’s

12. Example 3 All we have to do is choose which of the 7 blocks are going east School Harvey’s and which are going north

13. Example 3 We also could choose which of the 7 blocks are going north School Harvey’s and which are going east

14. Example 4 How many paths are there from school to Harvey’s if you can’t pass through X? X School Harvey’s

15. Example 4 Do it indirectly. # paths = total # paths – # paths that pass through X X School Harvey’s

16. Example 4 From school to X: 5 blocks (3 E, 2 N) so X School Harvey’s

17. Example 4 From X to Harvey’s: 2 blocks (1 E, 1 N) so X School Harvey’s

18. Example 4 # paths = total # paths – # paths that pass through X = X School Harvey’s