I am trying to design a dice game (up to 5 dice) and here is the problem I am looking to solve:
I am rolling $N$ dice, and I want to know the probability of rolling exactly $M$ ones (1's) & the total being $\geq K$.
Example:
I roll 4 dice, and I want to know what the probability of rolling exactly 1 one while the total sum of the dice is $\geq 10$.
There are $4$ possibilities for the die that shows the $1$. The other three dice cannot sum to $6,7,\text{ or }8.$ Since the $1$ doesn't appear on these dice the numbers must be anything but$$\begin{align} &2,2,2,\\&2,2,3,\\&2,3,3,\text{ or }\\&2,2,4\end{align}$$ which can occur in $1,3,3,\text{ or }3$ ways, respectively, for a total of $10$ ways. There are $5^3=125$ ways to roll three numbers greater than $1$, so we have $4\cdot115=460$ acceptable rolls. The probability of success is $$\frac{460}{6^4}=\frac{460}{1296}$$
The best way to approach the general problem is through the use of generating functions, but that's probably overkill for your application, since you won't, I imagine, be using more than a few dice. If that's true, the approach exemplified above should suffice.