  Nuclear structure at short $NN$-distances is still poorly understood. In
particular, the full quantum structure of the nucleus with a correlated
$NN$-pair is a challenge to theory. So far, model descriptions have been
limited to the average mean-field picture of the remaining nuclear system after
removing the $NN$-pair. In the recent experiment of the BM@N Collaboration at
JINR \cite{Patsyuk:2021fju}, the reactions
$^{12}\mbox{C}(p,2pn_s)^{10}\mbox{B}$ and
$^{12}\mbox{C}(p,2pp_s)^{10}\mbox{Be}$ induced by the hard elastic $pp$
scattering were studied. Here, $n_s$ or $p_s$ denote the undetected slow
nucleon in the rest frame of $^{12}\mbox{C}$. In contrast to the previous
experiments, the residual bound nucleus was also detected which requires a new
level of theoretical understanding. In the present work, we apply the technique
of fractional parentage coefficients of the translationally-invariant shell
model (TISM) to calculate the spectroscopic amplitude of the system $NN-B$
where $B$ is the remaining nuclear system. The spectroscopic amplitude enters
the full amplitude of a nuclear reaction. The relative $NN-B$ wave function is
no longer a free parameter of the model but is uniquely related to the internal
state of $B$. The interaction of the target proton with the $NN$-pair is
considered in the impulse approximation. We also include the initial- and final
state interactions of absorptive type as well as the single charge exchange
processes. Our calculations are in a reasonable agreement with the BM@N data.
