Supposing $i\in\{1,2,3,,\dots,m\}$, $r_i\in\{\beta_1, \beta_2,\dots,\beta_r\}$ and $\alpha_{ij}\in\{0,1,2,3,\dots,\alpha\}$ how many distinct monomials $\sum_{j=1}^p\prod_{i=1}^mr_i^{\alpha_{ij}}$ contribute to this sum if $p$ is fixed?
Each vector $(\alpha_{1j},\alpha_{2j},\dots,\alpha_{mj})$ at a particular $j$ is distinct.
Is it $\binom{(r(\alpha+1))^m}{p}$?
The monomials come from $\prod_{i=1}^mr_i^{\alpha_{ij}}$ at a given $j$. Each $j$ can give you a different monomial. There are $p$ choices of $j$. There are many more possible monomials, but given your $r$'s and $\alpha$'s you get only one per value of $j$.