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The example

I have the matrix in the image for a reference.

I can't understand why using combinations work on finding all the minimal paths. (In the image, I go from bottom-left to top-right).

For the first matrix we use Combinations of 2 elements taken once, but who those 2 elements are?? (Are they the right-arrows and up-arrows meaning).

Same for the 4 by 4 matrix. Do the elements I'm arranging look like those I put below the matrix? (right1, right2, up1,up2).

Example of a correct minimal path:1 2 5 8 9 or 1 2 5 4 9 Wrong would be: 1 2 5 6 7 8 9 or 1 2 3 2 5 8 9

I'm not sure it made sense, if not, please just explain however you can...

The Kebap
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    Can you explain the problem ? A bunch of matrices and mathematical notations is not a valid mathematical statement... Also, you can use mathjax to format mathematics. – caduk Mar 19 '24 at 09:03
  • I have to find all the minimal paths from 1 to 9 In the second matrix. This is just an example in order to understand how to use Combinatorics to find all the paths... – The Kebap Mar 19 '24 at 09:14
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    You have to edit the question to make it clear. Also, I still don't get the question. What is a path in a matrix, what is the length of a path, how does it relates to your notations, etc... – caduk Mar 19 '24 at 09:17
  • I added two examples of minimum lenght paths from 1 to 9. The notations in the parantheses would be the elements from which the paths are formed using combinations or permutations, as an example, one permutation would be (right, right, up, up) or (right, up, right, up). What I don't understand is why combinations of 4 elements taken 2 gives us the minimal paths – The Kebap Mar 19 '24 at 09:30
  • Because every minimal path is 4 steps, of which two steps are to the right and two steps are upwards, and those steps can be in any order. – Jaap Scherphuis Mar 19 '24 at 09:33
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    See one of these https://math.stackexchange.com/questions/636128/calculating-the-number-of-possible-paths-through-some-squares, https://math.stackexchange.com/questions/2829789/combinatorics-how-many-possible-paths – caduk Mar 19 '24 at 09:35
  • Yeah, I finally understood. Thank you guys – The Kebap Mar 19 '24 at 09:49

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