Fermi-Dirac Distribution constant

I was wondering about the constant in front of equation (2.2) in A and M and its origin. Shown below:

The rest of the equation is probability of finding an electron in a state with the velocity v. So then,

must be the amount of states per volume within the velocity range. To derive this consider
,
and then rearranging:

The lambda terms express the volume occupied by one electron with this state. So this is the amount of electrons that can fit into one unit volume with the given velocities.
And since electrons are spin half particles we can fit double the electrons into any volume. So the origins of C are demystified.