OPEN
Is it true that, for every integer $t\geq 1$, there is some integer $a$ such that
\[\binom{n}{k}=a\]
(with $1\leq k\leq n/2$) has exactly $t$ solutions?
Erdős
[Er96b] credits this to himself and Gordon 'many years ago', but it is more commonly known as
Singmaster's conjecture. For $t=3$ one could take $a=120$, and for $t=4$ one could take $a=3003$. There are no known examples for $t\geq 5$.
Both Erdős and Singmaster believed the answer to this question is no, and in fact that there exists an absolute upper bound on the number of solutions.
Matomäki, Radziwill, Shao, Tao, and Teräväinen [MRSTT22] have proved that there are always at most two solutions if we restrict $k$ to
\[k\geq \exp((\log n)^{2/3+\epsilon}),\]
assuming $a$ is sufficiently large depending on $\epsilon>0$.
