A consequence of the Gessel-Viennot-Lindstrom lemma is that if $A$ and $B$ are collections of points in $\mathbb{Z}^2$, that the number of non-intersecting lattice paths (counted with a sign) from the points in $A$ to the points in $B$ is given by the determinant of the matrix $M$ with $M_{(i,j)} = $ number of paths from $A_i$ to $B_j$ using only north and east unit steps in $\mathbb{Z}^2$. Note that these entries are binomial coefficients.
I am aware that this lemma is much more general. But nonetheless, it is a statement about a combinatorial interpretation of the determinant of a matrix of binomial coefficients.
Is there an analogous combinatorial interpretation of the hyperdeterminant of a tensor of multinomial coefficients.
With an example: the matrix $M_{(i,j)} = {a_i+b_j \choose a_i}$ has a determinant which is a count of non-intersecting lattice paths by Gessel-Viennot-Lindstrom.
Does the tensor $T_{(i,j,k)} = {a_i+b_j+c_k \choose a_i,b_j,c_k}$ have a hyperdeterminant which admits a combinatorial interpretation?