Counting the multiplicity of weight of a Verma module without using the Kostant partition function or the Weyl character formula.

Question

For a semisimple Lie algebra $\mathfrak{g}$ and a Cartan subalgebra $\mathfrak{h}$, if $\mu \in \mathfrak{h}$ and $W_{\mu}$ is the associated Verma module, I want to see that the multiplicity of $\lambda$ is equal to the number of ways $\mu - \lambda$ can be written as a non-negative integer combination of positive roots. I know that there are various ways in literature to do it, some of which include the Kostant's partition function or the Weyl character formula. However, I would like to know if we can conclude the same without using these tools?