Probability of dice roll (board games)

Assume that we have $n$ six-sided dice. We will roll all $n$ dice. What is the probability of getting at least $r$ ones, $s$ twos, $t$ threes and $u$ fours? Number $6$ can be used instead of any of other values. Number $5$ is bad all the time.

Example: we roll $4$ dice. What is the probability of getting at least $2$ twos and $1$ ones? Six can be used as one or two so rolling:

$1, 2, 2, 3$ is ok,

$1, 2, 3, 6$ is ok,

$6, 6, 6, 4$ is ok

but $2, 2, 3, 4$ is bad.