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 $\begin{array}{c}\frac{B\left(B^{+}\to D^{\ast +}{D}^{-}{K}^{+}\right)}{B\left(B^{+}\to {\overline{D}}^0{D}^0{K}^{+}\right)}=0.517\pm 0.015\pm 0.013\pm 0.011,\\ \frac{B\left(B^{+}\to D^{\ast -}{D}^{+}{K}^{+}\right)}{B\left(B^{+}\to {\overline{D}}^0{D}^0{K}^{+}\right)}=0.577\pm 0.016\pm 0.013\pm 0.013,\\ \begin{array}{c}\frac{B\left(B^0\to D^{\ast -}{D}^0{K}^{+}\right)}{B\left(B^0\to D^{-}{D}^0{K}^{+}\right)}=1.754\pm 0.028\pm 0.016\pm 0.035,\\ \frac{B\left(B^{+}\to D^{\ast +}{D}^{-}{K}^{+}\right)}{B\left(B^{+}\to D^{\ast -}{D}^{+}{K}^{+}\right)}=0.907\pm 0.033\pm 0.014,\end{array}\end{array}$ $$ {\displaystyle \begin{array}{c}\frac{\mathcal{B}\left({B}^{+}\to {D}^{\ast +}{D}^{-}{K}^{+}\right)}{\mathcal{B}\left({B}^{+}\to {\overline{D}}^0{D}^0{K}^{+}\right)}=0.517\pm 0.015\pm 0.013\pm 0.011,\\ {}\frac{\mathcal{B}\left({B}^{+}\to {D}^{\ast -}{D}^{+}{K}^{+}\right)}{\mathcal{B}\left({B}^{+}\to {\overline{D}}^0{D}^0{K}^{+}\right)}=0.577\pm 0.016\pm 0.013\pm 0.013,\\ {}\begin{array}{c}\frac{\mathcal{B}\left({B}^0\to {D}^{\ast -}{D}^0{K}^{+}\right)}{\mathcal{B}\left({B}^0\to {D}^{-}{D}^0{K}^{+}\right)}=1.754\pm 0.028\pm 0.016\pm 0.035,\\ {}\frac{\mathcal{B}\left({B}^{+}\to {D}^{\ast +}{D}^{-}{K}^{+}\right)}{\mathcal{B}\left({B}^{+}\to {D}^{\ast -}{D}^{+}{K}^{+}\right)}=0.907\pm 0.033\pm 0.014,\end{array}\end{array}} $$