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### Path Possibilities

```
Date: 10/24/1999 at 14:06:20
From: Roger
Subject: Path Possibilities

I have counted and calculated so many times and I'm still baffled. How
many combinations of moves are there to get from point A to point B
in the figure below, assuming that you can only move right and up?
Also assume you are on the corners and not the lines. Point A is the
very lower left corner, and B is the very upper right corner.

_ _ _ _ B
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
_ _|_|_|_|_|
|_|_|
_|_|_|
|_|
A

I know that if you take the number of possibilities of each set of
squares and multiply them, then you get the solution. The small square
has 2 possibilities. The second set of squares I think has 6. That is
where I get lost. Thanks for your time!

Roger
```

```
Date: 10/24/1999 at 16:22:23
From: Doctor Anthony
Subject: Re: Path Possibilities

Call a horizontal step H and a vertical step V.

For the first stage you must take one of each, and you can arrange
the letters H, V in two ways.

For the second stage you must do 2 H's and 2 V's. You can arrange
these four letters in

4!
----- = C(4,2) = 6 ways
2! 2!

For the third stage you must do 4 H's and 4 V's. These eight letters
can be arranged in

8!
----- = C(8,4) = 70 ways
4! 4!

Therefore the total number of different paths is

2 x 6 x 70 = 840

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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