Distribution of Dice Sums via Probability Generating Functions

Let $X_1, X_2, \ldots, X_n$ be iid rolls of a fair $d$-sided die, so each $X_i$ is uniform on $\{1, 2, \ldots, d\}$. Let $S_n = X_1 + X_2 + \cdots + X_n$.

(a) Derive the PGF $G_{X_1}(s) = E[s^{X_1}]$ in closed form.

(b) Write the PGF of $S_n$ and use it to derive $E[S_n]$ and $\text{Var}(S_n)$.

(c) For $n = 3$ fair six-sided dice ($d = 6$), use the PGF to find $P(S_3 = 10)$.

(d) Explain how the coefficient-extraction approach relates to the classical stars-and-bars counting with inclusion-exclusion for this problem.