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  1. math.stackexchange.com

    When you are thinking in squares instead of points then there $6$ steps to be taken. $1$ of them downwards and $5$ to the right. So it just comes to electing exactly $1$ of the $6$ consecutive step to be the steps downwards. ... {\left\vert\, #1 \,\right\vert}$ In general we have steps to right and down. Tipically, this is a possible path ...
  2. polypad.amplify.com

    Paths on a Grid Paths on a Grid. Below is an 8 by 8 grid. Point A is on the square at the top left corner, and point B is on the square of the bottom right corner. If you can only move down and to the right, how many different paths exist between A and B? Here are three of the possible paths as example; Let's have a closer look at these paths:
  3. Consider a hypothetical path from one corner to the opposite one and cut the grid in half along the column of horizontal segments. Let's look at the places where the path crossed from the left to the right or from right to the left and call them _corridors_. Let's try to count paths that cross the middle column in exactly these corridors.
  4. betterexplained.com

    What else could "Find paths on a grid" represent? Trap platform: Let's say you're making a set of trapdoors 4 × 6, with only 1 real path through (the others drop you into a volcano). What are the chances someone randomly walks through? With a 4×6 it's 210, as before. With a 12×12 grid it's 24!/12!12! = 2.7 million paths, with only 1 correct one.
  5. math.stackexchange.com

    With simple I mean no point is visited twice and the path can only one step north, south, east and west each time. I assume that this number is not exactly computable, so I am also interested in some partial questions to this problem:
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  1. The solution to the general problem is if you must take $X$ right steps, and $Y$ down steps then the number of routes is simply the ways of choosing where to take the down (or right) steps. i.e.

    $$ \binom{X + Y}{X} = \binom{X + Y}{Y} $$

    So in your example if you are traversing squares then there are 5 right steps and 1 down step so:

    $$ \binom{6}{1} = \binom{6}{5} = 6 $$

    If you are traversing edges then there are 6 right steps and 2 down steps so:

    $$ \binom{8}{2} = \binom{8}{6} = 28 $$

    --Daniel

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