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If A(0, 0, 0) and B(2, 2, 2) are pointsin coordinate space, how many paths are from A to B that move from one lattice point to another in the positive x, y, z direction?

 Dec 24, 2018
 #1
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each step can be denoted as (1,0,0), (0,1,0), (0,0,1)

 

the sum of the steps equals (2,2,2)

 

you are not allowed to overshoot in any direction

 

It should be pretty clear we need to take 2 steps in each direction.

 

So the total number of paths is just

 

\(N=\dbinom{6}{2}\dbinom{4}{2} = 90\)

 

The first term represents choosing two easts say, from the 6 possible steps.

The second term represents choosing two norths from the remaining 4 steps.

Give those the two up steps are fixed.

 

It doesn't matter which direction you pick for choosing first and second.  The results are the same.

 Dec 24, 2018
edited by Rom  Dec 24, 2018

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