Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be written as the sum of two $n$-polytopic numbers.
The above equation has infinitely many solutions for $n=1$ (trivial) and $n=2$ (the triplets $(a,b,c) =$ $(3,3,4)$, $(4,6,7)$, $(6,7,9)$, $(5,10,11)$, $(7,10,12)$, etc.). Also, regardless the value of $n$, it allows for at least one solution $(2n-1,2n-1,2n)$.
Is it true that for any $n$ the above equations has infinitely many solutions? Are there infinitely many $n$th order Binomial triplets?
[Edit]It appears this problem is known as Bombieri's Napkin Problem.[\Edit]