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The question is:

A partical moves in a the xy plane according to the following moves:

U :(m, n) → (m + 1, n + 1) (1)

L :(m, n) → (m + 1, n − 1), (2)

How many paths from (5, 5) to (30, 10) never touch or cross x−axis?

My reasoning is that we should calculate all the paths that touch or cross the x-axis, and substract the number from the total paths. There are total 25 steps, and there are 15 U's and 10 L's. I have divided this into different cases where the paths will touch the x-axis:

If the path begins with LLLLL

If the path has 6 L's and 1 U as the first 7 steps.

If the path has 7 L's and 2 U as the first 9 steps.

If the path has 8 L's and 3 U as the first 11 steps.

If the path has 9 L's and 4 U as the first 13 steps.

If the path has 10 L's and 5 U as the first 15 steps.

I have added all the cases and the number of paths that I got is 83 682. I know that I need to use the inclusion-exclusion principle since there will be duplicates. How do I use the principle and substract the duplicates?

Abhinav Kalra
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  • The linked duplicate does not exactly answer your question, but it shows the method you need to use. To count the "bad" paths which cross the $x$-axis at some point, use the reflection operation to relate these to paths from $(5,5)$ to a different final destination. – Mike Earnest Sep 06 '23 at 17:51
  • If there was no rule about the $x$-axis, the answer would be ${25 \choose 15}=3,268,760$. With the rule, it is slightly less, but still much larger than your $83,682$. – Henry Sep 06 '23 at 19:03

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