We know by "Fermat's Last Theorem" that the equation $$a^n + b^n = c^n$$ has no integer solutions $(a, b, c)$ for $n > 2$ other than when one of $a, b, c = 0$. However, the binomial coefficient analogue $$\binom{a}{n} + \binom{b}{n} = \binom{c}{n}$$ does apparently have nontrivial solutions. In the case $n = 3,$ for instance, MathPages gives a list of solutions to $x^3 - x + y^3 - y = z^3 - z$, and any such $x, y, z$ corresponds to a solution of the binomial coefficient equation above via $a = x+1$, $b = y+1$, $z = c+1$.
My question is both, if there's a parametrization of solution triples $(a, b, c)$ known for fixed or general values of $n$, and if there's a nice combinatorial interpretation of these triples. It seems like I could interpret this as, "The total number of ways to choose $n$ objects either out of a group of $a$ objects or out of a group of $b$ objects, equals the total number of ways to choose $n$ objects out of this other group of $c$ objects," but I'm having trouble in translating that into a parametrization or conditions on $a, b, c$. If anyone has references, I'd be interested in reading more on this.
I'm going to copy-paste the answer I posted ten years ago to a closely related question on MathOverflow. The link to MO is in my comment on the question here, and you will find other answers there, and many useful comments.
Some solutions for $n=3$ can be found at http://www.oeis.org/A010330 where there is also a reference to J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780, MR 19, 837f (but from the review it seems that paper deals with ${x\choose n}+{y\choose n}={z\choose n}+{w\choose n}$).
There are some other solutions at http://www.numericana.com/fame/apery.htm
EDIT Here are some more references for $n=3$:
Andrzej Krawczyk, A certain property of pyramidal numbers, Prace Nauk. Inst. Mat. Fiz. Politechn. Wrocƚaw. Ser. Studia i Materiaƚy No. 3 Teoria grafow (1970), 43--44, MR 51 #3048.
The author proves that for any natural number $m$ there exist distinct natural numbers $x$ and $y$ such that $P_x+P_y=P_{y+m}$ where $P_n=n(n+1)(n+2)/6$. (J. S. Joel)
M. Wunderlich, Certain properties of pyramidal and figurate numbers, Math. Comp. 16 (1962) 482--486, MR 26 #6115.
The author gives a lot of solutions of $x^3+y^3+z^3=x+y+z$ (which is equivalent to the equation we want). In his review, S Chowla claims to have proved the existence of infinitely many non-trivial solutions.
W. Sierpiński, Sur un propriété des nombres tétraédraux, Elem. Math. 17 1962 29--30, MR 24 #A3118.
This contains a proof that there are infinitely many solutions with $n=3$.
A. Oppenheim, On the Diophantine equation $x^3+y^3+z^3=x+y+z$, Proc. Amer. Math. Soc. 17 1966 493--496, MR 32 #5590.
Hugh Maxwell Edgar, Some remarks on the Diophantine equation $x^3+y^3+z^3=x+y+z$, Proc. Amer. Math. Soc. 16 1965 148--153, MR 30 #1094.
A. Oppenheim, On the Diophantine equation $x^3+y^3-z^3=px+py-qz$, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 230-241 1968 33--35, MR 39 #126.