Math Is Fun Forum

Melllyy

Guest

Pascals triangle

(click on the link to see the image)  http://mstevt0333.piczo.com

How many routes are there from a to b (southerly and easterly moves are only permitted)

- you do not need to draw each possible route
- consider the number of routes to each junction in turn, starting with those nearest to A
- look for some sort of relationship with Pascals triangle

Can any one explain what to do
thanks =]

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: Pascals triangle

By inspection I found that there are 10 routes if you can only move south and east. However, I wouldnt know how to help you with the use of Pascal's triangle.

Melllyy

Guest

Re: Pascals triangle

ok but does anyone know how i can find 10 by considering the number of routes to each junction in turn??

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Pascals triangle

The hints in your question are good ones. You have this picture:

x --- x --- x --- x
|     |     |     |
x --- x --- x --- x
|     |     |     |
x --- x --- x --- x

Now we replace each x with numbers to indicate how many ways there are of getting to each point.
The first one is easy - there's only one way of staying where you are and that's to not move.

1 --- x --- x --- x
|     |     |     |
x --- x --- x --- x
|     |     |     |
x --- x --- x --- x

Now consider the two points next to A. Because only east and south movement is allowed, you can only go to the square east of A by moving 1 east and you can only go to the square south of A by moving 1 south.

1 --- 1 --- x --- x
|     |     |     |
1 --- x --- x --- x
|     |     |     |
x --- x --- x --- x

Now it gets a bit more interesting. The only way to get to 2E is to move east from 1E. We've found that there's one way to get to 1E, which means that there's one way to get to 2E.
A similar argument applies for 2S.
But for 1S1E, you can either move south from 1E, or east from 1S. There's one way of getting to each of those, so there are two ways of getting to 1S1E.

1 --- 1 --- 1 --- x
|     |     |     |
1 --- 2 --- x --- x
|     |     |     |
1 --- x --- x --- x

Continue like that, filling in a diagonal with each step until you fill in the grid.

The relationship with Pascal's triangle might be easier to see if you rotate the completed grid 45° clockwise.

---

Why did the vector cross the road?
It wanted to be normal.

Melllyy

Guest

Re: Pascals triangle

thank you sooooo much