Equation for the multiplicity of a set of planes

The multiplicity (m) of lattice planes counts the number of planes related to (hkl) by symmetry. For example, the multiplicity of the {100} planes would be 6 because the following planes are all identical for a cubic crystal system;

begin{equation}
(100) : (010) :(001) :(bar100):(0bar10):(00bar1)
end{equation}

For planes not involving 0 such as {hkl}, {hhk} or {hhh} I think the formula below based on permuting h, k and l as entities and correcting for repetitions works:

begin{equation}
m=frac{3!times 2^3}{n!}
end{equation}

where n represents the number of repetitions (i.e n=0 for {hkl}, 2 for {hhk} and 3 for {hhh}).

How can I generalise this for cases involving zeros? The problem I see is that: begin{equation}
0=bar0
end{equation}

Below is a table showing some more values for different sets of planes: