How can I estimate $$ \Pr\left(f(x)=\sum_{i=0}^na_ix^i\text{ factors over }\mathbb{Q} \ \Bigg| \ \sum_ia_i=N, \ 0\leq a_i\in\mathbb{Z}\right)? $$ How about for fixed $n$ and $N\to\infty$?
The motivation is trying to factor $N=f(1)$ by randomly factoring $f$ from such a family, but I think the question (or related questions, say over other families) is interesting more generally.