Is there a simple form for the probability of rolling at least $n$ on $k$ 6-sided dice? Of course you can do it by recursion (see here). But is there a way to do it with just a few binomial coefficients, without the recursion?
What you want to know is here.
The formula for a homogeneous and fair dice from 1 to D values is this
where S is the sum that you want, n is the number of dice you roll and D is the number of sides of the dice (D6, D8, etc.)
To know the sum of at least S just need sum what you want over S.