In the role playing game Exalted, there is a dice mechanic whereby you have a certain number of 10-sided dice in a dice pool and when you roll them, each die showing a 7, 8, or 9 count as one success and those showing a 10 count as 2 successes (the idea being that you need some threshold number of successes to accomplish a task).
It's been a very long time since I've done this, but I do remember the basics of using generating functions to calculate these sorts of things. Specifically, I can write a polynomial representing a 10 dice pool as $(6 + 3x + x^2)^{10}$ and get Wolfram Alpha to do the heavy lifting for me to calculate the specific coefficients.
But then I got to wondering if there was a more general way to calculate the coefficient for all powers of x for an arbitrary n (i.e. for any number of dice in the pool). Moreover, it would be nice to know what the chance of getting at least k successes in a pool of size n would be, though I suspect that the answer to the former question leads immediately to the latter.