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The partial spatial separation of cold dark matter (DM) and gas is a ubiquitous feature in the formation of cosmic large-scale structure. This separation, termed dissociation, is prominent in galaxy clusters that formed through collisions of massive progenitors, such as the famous `Bullet' cluster. A direct comparison of the incidence of such dissociated structures with theoretical predictions is challenged by the rarity of strongly dissociated systems and the difficulty to quantify dissociation. This paper introduces a well-defined dimensionless dissociation index $S\in[-1,1]$ that encodes the quadrupole difference between DM and gas in a custom region. Using a simulation of cosmic large-scale structure with cold DM and ideal non-radiating gas, in $Λ$CDM cosmology, we find that 90 per cent of the haloes are positively dissociated ($S>0$), meaning their DM is more elongated than their gas. The spatial density of highly dissociated massive structures appears consistent with observations. Through idealised $N$-body+SPH simulations of colliding gaseous DM haloes, we further explore the details of how ram-pressure causes dissociation in binary collisions. A suite of 300 such simulations reveals a scale-free relation between the orbital parameters of binary collisions and the resulting dissociation. Building on this relation, we conclude that the frequency of dissociated structures in non-radiative cosmological simulations is nearly fully accounted for by the major (mass ratio $>1:10$) binary collisions predicted by such simulations. In principle, our results allow us to constrain the orbital parameters that produced specific observed dissociated clusters.