Grid Path problem

grid_path.php

<?php

require_once __DIR__ . "/include.php";

// This  shows how many ways we can go from the top left corner to the bottom right corner in a M x N grid
function grid_paths($m, $n)
{
    // Use memoization to avoid repeated calculations
    $memo = [];

    for($i = 0; $i < $m; $i++)
    {
        for($j = 0; $j < $n; $j++)
        {
            if($i == 0 || $j == 0)
            {
                $memo[$i][$j] = 1;
            }
            else
            {
                $memo[$i][$j] = $memo[$i - 1][$j] + $memo[$i][$j - 1];
            }
        }
    }

    return $memo[$m - 1][$n - 1];
}

// The mathematically correct way to do this is to use the formula (m + n - 2)! / (m - 1)! (n - 1)!
function grid_paths_math($m, $n)
{
    $factorial = function($n) {
        $result = 1;

        for($i = 1; $i <= $n; $i++)
        {
            $result *= $i;
        }

        return $result;
    };

    return $factorial($m + $n - 2) / ($factorial($m - 1) * $factorial($n - 1));
}

$m = 5;
$n = 4;

echo grid_paths($m, $n);
echo "\n";
echo grid_paths_math($m, $n);