How to calculate probability distribution of rolling n dice? (with a twist!) I'm trying to work on a project in code that asks this question. I'm currently able to get the right answer by breaking it up, but I'm wondering if there's a mathematical formula I can address this problem to as I'm actually facing computational resource limitations. 
For example, the amount of combinations for 50 dice is 50^6, which is 15,625,000,000 which is something my computer can't handle. 
I'm trying to find the probability distribution of getting a certain value of n dice where no single value of any dice is 1. 
For example, with a single dice, the probability of getting a 1,2,3,4,5, or 6 is 1/6.
For two dice, the probability of getting a total value of 4 or 12 is 1/36 (I ignore the case of 2 and 3 since one of the dice has to have a value of 1). The probability of a 5 or 11 WITHOUT a dice having a value of one is 1/18, 6 or 10 WITHOUT a dice having a value of 1 is 1/12, etc. etc.
Is anyone familiar with generating the probability distribution that all the possible values that can be rolled with n dice such that no dice contains a 1?
 A: Solve the problem for fifty 5-sided dices (and add 50 to the dice rolls).
You could do the first part (the standard dice roll problem) by 


*

*approximation with a normal distribution ( https://math.stackexchange.com/questions/406192/probability-distribution-of-rolling-multiple-dice ) 

*by the formula for the exact solution from Whuber mentioned in the comments.

*or by computation doing a convolution 49 times
rolls <- rep(0,5*50)

rolls[1:5] <- 1

for (i in 1:49) {
    rolls <- c(0,rolls[1:245],0,0,0,0) +
             c(0,0,rolls[1:245],0,0,0) +
             c(0,0,0,rolls[1:245],0,0) +
             c(0,0,0,0,rolls[1:245],0) +  
             c(0,0,0,0,0,rolls[1:245])
} 

The probability for 5-sided dice roll being x is rolls[x]/5^50. The probability for a 6-sided dice roll being x (and none of the dices is a one) is rolls[x-50]/5^60. Or rolls[x-50]/5^50 if you wish to condition on none of the dices being 1.
The above code can be improved by using 128 bit integers (if you find that necessary)
