Denis, including yours, this is a third solution:
(3*3!)(3*(3!)!/(3!) - 3*3) - 3! + 3/3 = 6313
________________________________________
(3*6)(3*6!/6 - 9) - 6 + 1 =
(18)(3*720/6 - 9) - 5 =
(18)(2160/6 - 9) - 5 =
(18)(360 - 9) - 5 =
(18)(351) - 5 =
6318 - 5 =
6313