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Paths on a Checkerboard


Date: 04/18/99 at 19:24:14
From: mike
Subject: Checkerboard

On an 8x8 checkerboard, how many paths are there from point A to point 
B (see the picture of the checkerboard shown below) following only the 
lines going downward and to the right?

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


Date: 04/19/99 at 07:33:47
From: Doctor Anthony
Subject: Re: Checkerboard

To go to the bottom righthand corner of the top block we must take 4 
horizontal steps and 4 vertical steps, in some order. This can be done 
in the same number of ways as arranging 4 H's and 4 V's in any order.  
That is 8 steps in all, 4 being alike of one kind and 4 alike of a 
second kind.

                    8!
Number of ways  = ------  =  70
                  4! 4!

For the single square in the middle there are 2 possible routes.

For the lower block with 3 H's and 3 V's the number of arrangements

         6!
     = ------  = 20
        3! 3!

Therefore the total number of routes = 70 x 2 x 20  =  2800

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

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