Neutrino reaction cross-sections, $( u_mu,mu^-)$, $( u_e,e^-)$, $mu$-capture and photoabsorption rates on $^{12}$C are computed within a large-basis shell-model framework, which included excitations up to $4hbaromega$. When ground-state correlations are included with an open $p$-shell the predictions of the calculations are in reasonable agreement with most of the experimental results for these reactions. Woods-Saxon radial wave functions are used, with their asymptotic forms matched to the experimental separation energies for bound states, and matched to a binding energy of 0.01 MeV for unbound states. For comparison purposes, some results are given for harmonic oscillator radial functions. Closest agreement between theory and experiment is achieved with unrestricted shell-model configurations and Woods-Saxon radial functions. We obtain for the neutrino-absorption inclusive cross sections: $bar{sigma} = 13.8 times 10^{-40}$ cm$^2$ for the $( u_{mu},mu^{-})$ decay-in-flight flux in agreement with the LSND datum of $(12.4 pm 1.8) times 10^{-40}$ cm$^2$; and $bar{sigma} = 12.5 times 10^{-42}$ cm$^2$ for the $( u_{e},e^{-})$ decay-at-rest flux, less than the experimental result of $(14.4 pm 1.2) times 10^{-42}$ cm$^2$.
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