I gave a short answer to this question here: https://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators.
A more general answer says the following: if $K$ is a field, $R$ is an $\mathbb{N}$-graded $K$-algebra finitely generated over $K$, and $M$ a $\mathbb{Z}$-graded finitely generated $R$-module, then
$$\beta_{ij}(M)=\dim_K\operatorname{Tor}_i^R(K,M)_j,$$
where $\beta_{ij}(M)$ are the graded Betti numbers of $M$.
