TITLE: A binomial generalization of the FLT: Bombieri's Napkin Problem
QUESTION [35 upvotes]: 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?
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 OEIS link 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. Bremner, Duke Math. J. 44 (1977) 757--765].
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.
Leech's paper indicates
the particular solution
$$
\binom{132}{4}+\binom{190}{4}=\binom{200}{4}
$$
and the trivial infinite family
$$
\binom{2n-1}n+\binom{2n-1}n=\binom{2n}n.
$$

REPLY [7 votes]: My first instinct is to say it seems unlikely there's been serious progress on this problem for general n.  Unlike the Fermat equation, this one is not homogeneous of degree n, which means that it's really a question about points on a surface rather than points on a curve.  We don't have a giant toolbox for controlling rational or integral points on surfaces as we do for curves.
In fact, I can't think of any example of a family of surfaces of growing degree where we can prove a theorem like "there are no nontrivial solutions for n > N."  OK, I guess one knows this about the symmetric squares of X_1(n) by Merel...