  Let $\mathbb{W}$ be an irreducible subvariety a Hilbert scheme $Hilb_{p_W}
(t) (\mathbb{P}^n )$. We show that under mild hypothesis there are polynomial
formulas for the degrees of the loci of hypersurfaces in $\mathbb{P}^n$ with
singular subschemes containing some member of the family $\mathbb{W}$. The
formulas are made explicit in a number of cases.
