4.2 Pascal’s Triangle

Consider the binomial expansions…

0( ) 1x a 1( )x a x a

2 2 2( ) 2x a x ax a 3 3 2 2 3( ) 3 3x a x x a xa a

( ) ?nx a

Let’s look at the coefficients…1

1 11 2 1

1 3 3 14( )x a 1 4 6 4 15( )x a 1 5 10 10 5 1

Pascal’s Triangle

( ) ?nx a

1

1 11 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1Etc.

Pascal’s Triangle and Paths…

1

1 11 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

How many paths can you take to get to the indicated point?

22 ways

How many paths can you take to get to the indicated point?

Pascal’s Triangle and Paths…

1

1 11 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 14

4 ways

Pascal’s Triangle and Paths…

1

1 11 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

How many paths can you take to get to the indicated point?

10

10 ways

1

1 way

10

Example 1Determine 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 AS S S

C C C CA A A

L L

Write the triangle coefficients

1

1 1

1

2

1 133

4 6 4

10 10

There are 10+10 = 20 paths that will spell PASCAL.

Example 2On 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 3How 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 3Note: there are 7 blocks in total to travel:4 going east,

School

Harvey’s

3 north.

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

School

Harvey’s

Example 3All we have to do is choose which of the 7 blocks

are going east

School

Harvey’s7

4

and which are going north3

3

35

Example 3We also could choose which of the 7 blocks are

going north

School

Harvey’s7

3

and which are going east4

4

35

Example 4How many paths are there from school to

Harvey’s if you can’t pass through X?

X

School

Harvey’s

Example 4Do it indirectly.

# paths = total # paths – # paths that pass through X

X

School

Harvey’s

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

X

School

Harvey’s

5

3

2

2

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

X

School

Harvey’s

5

3

2

2

2

1

1

1

Example 4# paths = total # paths – # paths that pass through X

=

X

School

Harvey’s

7

4

3

3

15

5

3

2

2

2

1

1

1