  In this paper, binary systems of unequal Kerr-Newman black holes located on
the axis and apart by a massless strut are investigated. After adopting a
fitting parametrization, the conditions on the axis and the one eliminating the
individual magnetic charges are solved exactly with the aim to obtain a
7-parametric asymptotically flat exact solution. It is also deduced explicit
analytical formulas for each half-length horizon ${\sigma}_{i}$, $i = 1,2$, in
terms of arbitrary physical Komar parameters: mass $M_{i}$, electric charge
$Q_{i}$, angular momentum $J_{i}$, and a coordinate distance $R$. All the
thermodynamical properties for the black holes as well as the interaction force
related to the strut are also given in a concise way. Our analysis permits us
to derive in detail the physical limits of the solution and introduce some
scenarios where the strut is absent during the merger process.
