This was one of the mensa puzzles in a box that I have.

possibleRoutesThroughGrid.php

<?php

function factorial($number)
{
    $factorial = $number;
    $i = $number - 1;
    while ($i > 0) {
        $factorial *= $i--;
    }
    return $factorial;
}

function possibleRoutes($rows, $columns)
{
    $m = $rows - 1;
    $n = $columns - 1;
    $routes = factorial($m + $n) / (factorial($m) * factorial($n));
    return $routes;
}

assert(possibleRoutes(3, 3) == 6);
assert(possibleRoutes(4, 3) == 10);
assert(possibleRoutes(6, 6) == 252);