This is an extract from <a href="http://peccatte.karefil.com/PhiMathsTextes/Apery.html">Apéry's biography</a>
(which some of the people have already enjoyed in
<a href="http://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty/25631#25631">this answer</a>).

> 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?

**Edit.** I owe the following official name of the problem to Gerry,
as well as Alf van der Poorten's (different!) point of view on this story and
some useful links on the problem (see Gerry's comments and response).
The name is *Bombieri's Napkin Problem*. As the <a href="http://www.oeis.org/A010330">OEIS link</a> suggests,
Bombieri said that
  
> "the equation $\binom xn+\binom yn=\binom zn$
> has no *trivial* solutions for $n\ge 3$"

(the joke being that he said "trivial" rather than "nontrivial"!).

As Gerry indicates in his comments, the special case $n=3$ has a long history
started from the 1915 paper [Bökle, *Z. Math. Naturwiss. Unterricht* **46** (1915), 160];
this is reflected in
<a href="http://dx.doi.org/10.1215/S0012-7094-77-04433-7">[A. Bremner, *Duke Math. J.* **44** (1977) 757--765]</a>.
A related link is [F. Beukers, *Fifth Conference of the Canadian Number Theory Association*, 25--33]
for which I could not find an MR link.
<a href="http://dx.doi.org/10.1017/S0305004100032850">Leech's paper</a> indicates
the particular solution
$$
\binom{132}{4}+\binom{190}{4}=\binom{200}{4}
$$
and the trivial infinite family
$$
\binom{2n-1}n+\binom{2n-1}n=\binom{2n}n.
$$