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

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Presentation transcript:

4.2 Pascal’s Triangle

Consider the binomial expansions… Let’s look at the coefficients…

Pascal’s Triangle Etc.

Pascal’s Triangle and Paths… How many paths can you take to get to the indicated point? 2 2 ways

How many paths can you take to get to the indicated point? Pascal’s Triangle and Paths… ways

Pascal’s Triangle and Paths… How many paths can you take to get to the indicated point? ways 1 1 way 10

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 There are = 20 paths that will spell PASCAL.

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.

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.

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

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

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

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

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

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

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

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

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