I'm trying to solve this problem, but I'm stuck establishing the distribution of $N$ to use the Fisher–Neyman factorization theorem.
Let $X$ take on the specified values $v_1, . . . , v_k$ with probabilities $θ_1, . . . , θ_k$, respectively. Suppose that $X_1, . . . , X_n$ are independently and identically distributed as $X$. Suppose that $θ = (θ_1, . . . ,θ_k)$ is unknown and may range over the set $Θ = \{(θ_1, . . . , θ_k) : θ_i ≥ 0, 1 ≤ i ≤ k, \sum_{i=1}^{k} θ_i = 1 \}$. Let $N_j$ be the number of $X_i$ which equal $v_j$ .
(a) What is the distribution of $(N_1, . . . , N_k)$?
(b) Show that $N = (N_1, . . . , N_{k-1})$ is sufficient for $θ$.
$(N_1, N_2, \ldots , N_k)$ has a multinomial distribution
so (with a slight abuse of notation) a probability of $$\frac{n!}{N_1! N_2! \ldots N_k!}\theta_1^{N_1!}\theta_2^{N_2!} \ldots \theta_k^{N_k!}$$
For (b), a first step would be showing $(N_1, N_2, \ldots , N_{k-1} , N_k)$ is sufficient for $\theta$ and then using $N_k = n-(N_1+N_2+\cdots + N_{k-1})$