A tongue-in-cheek recent question here inspired me another, less ambiguous question : say that a binom coefficient $\binom{x}{y}$ (where $x$ and $y$ are positive integers) is nontrivial-and-normalized if $1<y \leq \frac{x}{2}$. The question then goes, find all "coincidences" of the form $\binom{x}{y}=\binom{x'}{y'}$ where both $\binom{x}{y}$ and $\binom{x'}{y'}$ are nontrivial-and-normalized, and of course $(x,y)\neq (x',y')$. If this is too hard, does this equation have finitely or infinitely many solutions ?
My thoughts : with a computer, I found that the only coincidences with $\max(x,x')\leq 100$ are
$$ \begin{array}{c} 120=\binom{16}{2}=\binom{10}{3}, 210=\binom{10}{4}=\binom{21}{2}, \\ 1540=\binom{22}{3}=\binom{56}{2}, 3003=\binom{14}{6}=\binom{15}{5}=\binom{78}{2}. \end{array} $$
Update 05/26/2017: Judging from all the references given in the comments, the first question ("find all the coincidences") is an open problem, and the second question has an affirmative answer : there is indeed an infinite number of solution pairs. See the Wikipedia link on Singmaster's conjecture, quoted in the comments.