Let $k, \ell \geq 2$ be positive integers. Do there exist infinitely many positive integers $N$ such that the equation
$$\displaystyle \binom{m}{k} \binom{n}{\ell} = N$$
have more than four solutions? I am most interested in the case $k = \ell = 2$.
This is related to Singmaster's conjecture, which asks for solutions to
$$\displaystyle \binom{m}{k} = \binom{n}{\ell}$$
where $m,n,k,\ell$ are all variable. In the above question, we are considering fixed $k, \ell$.
