  In this paper, we solve a classical counting problem for non-degenerate forms
of symplectic and hermitian type defined on a vector space: given a subspace
$\pi$, we find the number of non-singular subspaces that are trivially
intersecting with $\pi$ and span a non-singular subspace with $\pi$. Lower
bounds for the quantity of such pairs where $\pi$ is non-singular were first
studied in ``Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)'', which was
later improved in ``Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)''
and generalised in ``Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)''.
In this paper, we derive explicit formulae, which allow us to give the exact
proportion and improve the known lower bounds.
