10 faced dice problem - for gamers -

[Guest]

Many rpg games have the D10 system, that work like this:
- set the difficulty level (K)
- roll a certain amount of d10 (N)
-> the success (S) consists of how many of those dices is equal or exceed K, but there are two important rules:
1) if you roll 1 you have to remove a success
2) if you roll 10 you can remove a 1

At the end of the roll, there are 4 possible state
1) S = 0 , some 1 -> critical failure
2) S = 0 -> normal failure
3) S >= 0 , no 10 -> normal success
4) S >= 0 , some 10 -> critical success

There's a mathematical formula that can calculate the S probability of every state (the state 3 and 4 are the most interesting)?
P(10,N,K)