Given n ∈ N, is there an m ∈ M s.t. m occurs exactly n times in Pascal's Triangle?

It seems that, given m, you can set up a system of equations like this: C(x₁, x₂) = C(x₃, x₄) = C(x₅, x₆) = … = m

Then it's just a matter of searching through 2*m-dimensional space for a solution to all these equations.

You can probably expand everything out into a horrifying system of polynomials and set up a Grobner basis..then somehow use that to extract a solution, or at least you could see if none exists.