Paths on a CheckerboardDate: 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/ |
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