Measurement of the branching fractions for $B^+\to D^{*+}D^-K^+$, $B^+\to D^{*-}D^+K^+$, and $B^0\to D^{*-}D^0K^+$ decays

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 $9.1$ fb$^{-1}$. The measured branching-fraction ratios are \begin{equation*} \begin{split} \frac{\mathcal{B} (B^+\to D^{*+}D^-K^+)}{\mathcal{B} (B^+\to \overline{D}{}^0D^0K^+)} &= 0.517 \pm 0.015 \pm 0.013 \pm 0.011 , \\ \frac{\mathcal{B} (B^+\to D^{*-}D^+K^+)}{\mathcal{B} (B^+\to \overline{D}{}^0D^0K^+)} &= 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^0K^+)} &= 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{split} \end{equation*} 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.