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<?php

/*****************************

 * ProjectEuler - Problem 15

 * By Ross Marks

 *****************************

 * Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.

 * 

 * How many such routes are there through a 20×20 grid?

 ****************************/

$no = 1;

$gridsize = 20;



for ($i = 0; $i < $gridsize; $i++){

  $no *= (2 * $gridsize) - $i;

  $no /= $i +1;

}



echo "result: $no";



/*

Notes:



Paths as a series of Ds and Rs. 

In 2×2 grid all paths are 1) DDRR 2) DRDR 3) DRRD 4) RDRD 5) RDDR 6) RRDD.



all paths = 2N containing N Rs & N Ds.

If we have 2N empty spaces and place all Rs, then the placement of the Ds are given

SO



how many ways can N out of 2N possible places?

*/



?>

Project Euler Solutions by Ross Marks