Combinatorics and Rolling Dice Similarity? Define a function $F(A, B, C)$ as the number of ways you can roll $B$ $C$-sided dice to sum up to $A$, counting different orderings (rolling a $2$, $2$, and $3$ with three dice is different from rolling a $2$, $3$, and $2$).
Example:
With three $5$-sided dice, the list of $F(A, B, C)$ values in the domain of the possible values of $A$ for $B = 3$ and $C = 5$ is:
$$F(3, 3, 5), F(4, 3, 5), F(5, 3, 5), F(6, 3, 5), ... , F(15, 3, 5)$$ is evaluated to:
$$1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1$$
Call this list $L_1$.
Let $s$ be the number of sides on each die, let $n$ be the number of dice, and let $v$ be the total value to roll from the $n$ dice.
Let $L_2$ be the list of ${v - 1}\choose{v - n}$ in the domain of $v$ values for $n = 3$.
Then $L_2$ is:
$${{3 - 1}\choose{3 - 3}}, {{4 - 1}\choose{4 - 3}}, {{5 - 1}\choose{5 - 3}}, {{6 - 1}\choose{6 - 3}}, ... , {{15 - 1}\choose{15 - 3}}$$
Which is evaluated to:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91$$
Comparing $L_1$ with $L_2$, we see that only the first $s$ values of the lists are equal:
$$1, 3, 6, 10, 15$$
I have observed that this property holds with other values of $s$, $v$, and $n$, and $A$, $B$, and $C$.
Can someone please explain why $L_1$ and $L_2$ share the first $s$ values?

Refer to answers to  Rolling dice problem , because this is the same as finding a $B$-tuple, with values in the range $1..C$, summing up to $A$, i.e. with values in the range $0..C-1$, summing up to $A-B$. So
$$N_{\,b} (s,r,m) = \text{No}\text{. of solutions to}\;\left\{ \begin{gathered}
  0 \leqslant \text{integer  }x_{\,j}  \leqslant r \hfill \\
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,m}  = s \hfill \\ 
\end{gathered}  \right.$$
with $m=B,\ r=C-1,\ s=A-B$.
The formula for $Nb(s,r,m)$ is given in the answers to the question linked above.
