This is an extract from Apéry's biography (which some of the people have already enjoyed in this answer).

During a mathematician's dinner in Kingston, Canada, in 1979, the conversation turned to Fermat's last theorem, and Enrico Bombieri proposed a problem: to show that the equation $$ \binom xn+\binom yn=\binom zn > \qquad\text{where}\quad n\ge 3 $$ has no nontrivial solution. Apéry left the table and came back at breakfast with the solution $n = 3$, $x = 10$, $y = > 16$, $z = 17$. Bombieri replied stiffly, "I said nontrivial."

What is the state of art for the equation above? Was it seriously studied?