The question is:
“In how many ways can you roll $3$ regular distinguishable dice, such that the sum of points on their top surface is 8?”
My answer is $(8+3-1,3-1)$/ $r=8$. But I listed all possible sums, it exactly equal to $21$, and also looked at the correct answer, which is $C(5+3-1,3-1)$. I don’t know why $r=5$, not $8$.
The $+3$ that you add to $8$ in your computation accounts for the summands being $0$, but there is no $0$ on the dice. You just have to split $8$ stars with two bars that you have $8-1$ places to put.