Binary-driven hypernovae (BdHNe) within the induced gravitational collapse paradigm have been introduced to explain energetic (E_{iso}≳10^{52} erg), long gamma-ray bursts (GRBs) associated with type Ic supernovae (SNe). The progenitor is a tight binary composed of a carbon-oxygen (CO) core and a neutron-star (NS) companion, a subclass of the newly proposed "ultrastripped" binaries. The CO-NS short-period orbit causes the NS to accrete appreciable matter from the SN ejecta when the CO core collapses, ultimately causing it to collapse to a black hole (BH) and producing a GRB. These tight binaries evolve through the SN explosion very differently than compact binaries studied in population synthesis calculations. First, the hypercritical accretion onto the NS companion alters both the mass and the momentum of the binary. Second, because the explosion time scale is on par with the orbital period, the mass ejection cannot be assumed to be instantaneous. This dramatically affects the post-SN fate of the binary. Finally, the bow shock created as the accreting NS plows through the SN ejecta transfers angular momentum, braking the orbit. These systems remain bound even if a large fraction of the binary mass is lost in the explosion (well above the canonical 50% limit), and even large kicks are unlikely to unbind the system. Indeed, BdHNe produce a new family of NS-BH binaries unaccounted for in current population synthesis analyses and, although they may be rare, the fact that nearly 100% remain bound implies that they may play an important role in the compact merger rate, important for gravitational waves that, in turn, can produce a new class of ultrashort GRBs.