You roll Nd6 (N six sided die). For each die with value M or over, you count 1 success.
For each die that rolls a six, roll another die.
How would you calculate the probability that N dice roll S successes?
Not sure how you mean that "for each die that rolls a six, roll another" line. Does that mean that for an N of 10, you might actually roll 11 or 12?
The probability of getting a success (P) is (7-M)/6.
The mean number of successes is PN.
The standard deviation of the number of successes over numerous trials is sqrt( P*(1-P)/N )
To find the probability of S successes, use the mean and standard deviation above and look up S on a normal distribution probability chart.
If you want to say that 10 rolls could actually mean 11 or 12 rolls, use N'=7N/6 in the equations.