  Using proton-proton collision data, corresponding to an integrated luminosity
of $9fb^{-1}$ collected with the LHCb detector, seven decay modes of the
$B_c^+$ meson into a $J/\psi$ or $\psi(2S)$ meson and three charged hadrons,
kaons or pions, are studied. The decays $B_c^+\rightarrow ( \psi(2S)
\rightarrow J/\psi \pi^+ \pi^-)\pi^+ $, $B_c^+\rightarrow
\psi(2S)\pi^+\pi^+\pi^-$, $B_c^+\rightarrow J/\psi K^+\pi^+\pi^-$ and
$B_c^+\rightarrow K^+K^+K^-$ are observed for the first time, and evidence for
the $B_c^+\rightarrow \psi(2S)K^+K^-\pi^+$ decay is found, where $J/\psi$ and
$\psi(2S)$ mesons are reconstructed in their dimuon decay modes. The ratios of
branching fractions between the different $B_c^+$ decays are reported as well
as the fractions of the decays proceeding via intermediate resonances. The
results largely support the factorisation approach used for a theoretical
description of the studied decays.
