As stated the first person that can help me answer this question I will give credit to them if and when I get the game I'm working on published!
I am using a pool of $10$-sided dice and counting a success when a die rolls a 9 or a 10. Each die acts independently of the others to achieve a success so that if you had a pool of $5$ dice, you could achieve $5$ successes. I need to know the probability of rolling $1$, $2$, or $3$ successes in a pool of dice of sizes ranging from $1$ to $12$. Obviously you have a $20\%$ chance to roll $1$ success with a dice pool of $1$. Sadly, that is about as far as my limitation of probability mathematics can get me.
Any help would be greatly appreciated!