public static function permutations(int $n, int $k = null)
{
if ($n < 0) {
throw new Exception\OutOfBoundsException('Cannot compute negative permutations.');
}
if (!is_null($k) && $k > $n) {
throw new Exception\OutOfBoundsException('k cannot be larger than n.');
}
$n！ = self::factorial($n);
// nPn: permutations of n things, taken n at a time
if (is_null($k)) {
return $n！;
}
// nPk: Permutations of n things taking only k of them
$⟮n − k⟯！ = self::factorial($n - $k);
return $n！ / $⟮n − k⟯！;
}