I wanted to leave this as a comment, but for some reason I can't.
I gave a short answer to this question here: http://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators.
A more general answer says the following: if $R$ is a graded $K$-algebra, $K$ a field, and $M$ a 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 $M$.