The rate for Higgs (H) bosons production in association with either one ($\mathrm{tH}$) or two (${\mathrm{t}}{\mathrm{\bar{t}}}\mathrm{H}$) top quarks is measured in final states containing multiple electrons, muons, or tau leptons decaying to hadrons and a neutrino, using proton-proton collisions recorded at a center-of-mass energy of 13 TeV by the CMS experiment. The analyzed data correspond to an integrated luminosity of 137 fb$^{-1}$. The analysis is aimed at events that contain $\mathrm{H} \to \mathrm{W}\mathrm{W}$, $\mathrm{H} \to \tau\tau$, or $\mathrm{H} \to \mathrm{Z}\mathrm{Z}$ decays and each of the top quark(s) decays either to lepton+jets or all-jet channels. Sensitivity to signal is maximized by including ten signatures in the analysis, depending on the lepton multiplicity. The separation among the $\mathrm{tH}$, the ${\mathrm{t}}{\mathrm{\bar{t}}}\mathrm{H}$, and the backgrounds is enhanced through machine-learning techniques and matrix-element methods. The measured production rates for the ${\mathrm{t}}{\mathrm{\bar{t}}}\mathrm{H}$ and $\mathrm{tH}$ signals correspond to 0.92 $\pm$ 0.19 (stat) $^{+0.17}_{-0.13}$ (syst) and 5.7 $\pm$ 2.7 (stat) $\pm$ 3.0 (syst) of their respective standard model (SM) expectations. The corresponding observed (expected) significance amounts to 4.7 (5.2) standard deviations for $\mathrm{t}\mathrm{\bar{t}}\mathrm{H}$, and to 1.4 (0.3) for $\mathrm{t}\mathrm{H}$ production. Assuming that the Higgs boson coupling to the tau lepton is equal in strength to its expectation in the SM, the coupling $y_{{\mathrm{t}}}$ of the Higgs boson to the top quark divided by its SM expectation, ${\kappa_{{\mathrm{t}}}}=y_{{\mathrm{t}}}/y_{{\mathrm{t}}}^{\mathrm{SM}}$, is constrained to be within $-0.9 < {\kappa_{{\mathrm{t}}}} < -0.7$ or $0.7 < {\kappa_{{\mathrm{t}}}} < 1.1$, at 95% confidence level. This result is the most sensitive measurement of the ${\mathrm{t}}{\mathrm{\bar{t}}}\mathrm{H}$ production rate to date.

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