  A measurement of four branching-fraction ratios for three-body decays of $B$
mesons involving two open-charm hadrons in the final state is presented. Run 1
and Run 2 $pp$ collision data are used, recorded by the LHCb experiment at
centre-of-mass energies $7$, $8$, and $13$ TeV and corresponding to an
integrated luminosity of $9$ fb$^{-1}$. The measured branching-fraction ratios
are \[ \begin{eqnarray} \frac{\mathcal{B} (B^+\to D^{*+}D^-K^+)}{\mathcal{B}
(B^+\to \kern 0.2em\overline{\kern -0.2em D}{}^0 D^0 K^+)} &=& 0.517 \pm 0.015
\pm 0.013 \pm 0.011 , \\ \frac{\mathcal{B} (B^+\to D^{*-}D^+K^+)}{\mathcal{B}
(B^+\to \kern 0.2em\overline{\kern -0.2em D}{}^0 D^0 K^+)} &=& 0.577 \pm 0.016
\pm 0.013 \pm 0.013 , \\ \frac{\mathcal{B} (B^0\to D^{*-}D^0K^+)}{\mathcal{B}
(B^0\to D^- D^0 K^+)} &=& 1.754 \pm 0.028 \pm 0.016 \pm 0.035 , \\
\frac{\mathcal{B} (B^+\to D^{*+}D^-K^+)}{\mathcal{B} (B^+\to D^{*-}D^+K^+)} &=&
0.907 \pm 0.033 \pm 0.014 ,\end{eqnarray} \] where the first of the
uncertainties is statistical, the second systematic, and the third is due to
the uncertainties on the $D$-meson branching fractions. These are the most
accurate measurements of these ratios to date.
