I have X dice with 6 sides each & want to simulate how many 6's I get by using ONE random number 0 to Y

Question

With 6 sides on, say, 3 dice, I have 216 combinations.

 

** Three 6's, 1 possibility: 666

** Two 6's, 15 possibilities: 665 664 663 662 661 656 655... etc

** One 6: 75 possibilities: 611 612 613 614 615 621 622 623... etc

** Zero 6's: 125 possibilities: 111 112 113 114 115 121 122 123 124 125... etc

 

 

Instead of rolling three 6-sided dice, I could roll one 216 sided die

If I roll a 1, it counts as three 6's

2-16 counts as two 6's

17-92 counts as one 6

93-216 counts as zero 6's

 

So here's where it gets tricky for me.  Lets say I have an arbitrarily large number of dice to roll.... say... 325 of them.  6^325 is an extremely big number to work with, and I'm unsure of how to calculate the ranges.

 

 

 

Answer 1

With n dice, we can count the ways to get k 6's as follows.

 

Choose which rolls are 6's: C(n,k) choices.

 

Choose value rolled by each of the other dice: 5 for each of n-k dice for a total of 5^n-k.

 

Thus there are 5^n-kC(n,k) ways to roll exactly k 6's

 

Now, the ranges can be calculated by summing ways to get j 6's for j=0,...,k.

 

Thus the range of X's in {1,2,3,...,6ⁿ} for k 6's is:

 

1+∑_j=1...k-1 5^n-jC(n,j) ≤ X ≤ ∑_j=1...k 5^n-jC(n,j).

 

Here, C(n,k) is the binomial coefficient, given by n!/[k!(n-k)!].