  Hadronic states composed of multi-quark flavors may exist in reality since
they are not prohibited by QCD. Compact four quark systems of color singlet are
classified as tetraquarks. To understand the properties of these states, more
theoretical and experimental efforts are needed. In this work, we study charmed
tetraquarks with three light flavors using flavor $SU(3)$ symmetry. States with
three different light quarks must be in a ${\bf \bar 6 }$ or a ${\bf 15}$
multiplet. We investigate the production of charmed tetraquarks $X_c$ in $B\to
X_c (\overline {X}_c) P$ and $B_c \to X_c P$ decays. Whether the states with
three light quarks belong to ${\bf \bar 6}$ or ${\bf 15}$ can be determined by
studying various tetraquark $B$ and $B_c$ decays. We demonstrate that the decay
amplitudes for these decays can be parametrized by a few irreducible SU(3)
invariant amplitudes. We then derive relations for decay widths and CP
violating rate difference which can be examined experimentally. Although no
experimental measurement is available yet, they might be accessed at the
ongoing and forthcoming experiments like the LHCb and Belle-II. Measurements of
these observables can not only provide useful information for the study of
exotics spectroscopy but are also valuable information towards a better
understanding of some non-perturbative aspects of QCD.
