Trichotomy theorem (elementary, complete proof in THEOREM-v2.md): every solution satisfies exactly one of (i) $P(\binom nk)\le n/2$ and $(n-k,n]$ contains no prime; (ii) $P>n-k$ and $(n/2,n-k]$ contains no prime; (iii) $\binom nk=\prod_{n-k<p\le n}p$ exactly. Horn (iii) is cornered: $n\le k^3$, $(\log k)(\log(n/k))\le 2\log n$ (Montgomery–Vaughan), hence $k>n/13.4$ for $n\ge 10^5$ and $k\ge(e^{-2}-o(1))n$; and horn (iii) is exhaustively empty for $n\le 10^5$ (all $k$). All nine known solutions are machine-verified to satisfy the trichotomy — six in horn (ii) at $n\le 15$, three in horn (i).