Erdős Problem 849

Reference: erdosproblems.com/849

theorem Erdos849.erdos_849 : sorry ↔ ∀ t ≥ 1, ∃ (a : ℕ), {n : ℕ | ∃ k ≥ 1, 2 * k ≤ n ∧ n.choose k = a}.ncard = t

Is it true that, for every integer $t\geq1$, there is some integer $a$ such that ${n \choose k} = a$ with $1\leq k \le \frac{n}{2}$ has exactly $t$ solutions?