I like how at the end of the article Bill Roberts posted it says " Now if you really want a challenge here is one to try. How many ways are there to roll a total of 15 with 5 dice? You should get an answer of 651."
And we’re looking for total ways to roll 20 with 5 dice.
Let me preface by saying, I may be totally off base here. This just popped in my head and I haven’t really worked it out yet.
But, if there are 651 different ways to roll 15 with 5 dice, I think there would also be 651 for 20 with 5 dice. My thinking here is that if you chart the combinations to make 20, your “base” (for lack of a better term) will be 44444, or a set of five fours, and every permutation “above it”.
The opposite would be 33333, a set of five threes and every permutation below it, which the guy in the article calculated to be 651.
What I mean by “above” and “below” is, if you take the guy’s charting idea from the article, you start with the 6 as the first die and chart the dice in descending order to make the desired total. For example, the first few combinations of 20 would be:
6 6 6 1 1
6 6 5 2 1
6 6 4 3 1
6 6 4 2 2
etc, etc until you reach 4 4 4 4 4
Now I believe, and this is where I’m just guessing, that you come out with the same amount of combinations and permeations for a total of 15.