The current limitations of the standard model of elementary particle physics drive the search for phenomenons forbidden within that framework with the goal to unveil a more universal theory. This thesis presents the search for two lepton flavour violating decays forbidden in the standard model: $B^{+} \rightarrow K^{+} \tau^{-} \mu^{+}$ and $B^{+} \rightarrow K^{+} \tau^{+} \mu^{-}$. The search is performed by studying proton-proton collision data collected by the detector of the LHCb collaboration corresponding to $9.1\text{fb}^{-1}$ of integrated luminosity. No significant signal is observed and as such the following upper limits at $90\%$ confidence level are set: $\mathcal{B}(B^{+} \rightarrow K^{+} \tau^{-} \mu^{+}) <1.3 \times 10^{-6}$ and $\mathcal{B}(B^{+} \rightarrow K^{+} \tau^{+} \mu^{-}) <1.2 \times 10^{-6}$. These limits correspond to one order of magnitude improvement relatively to the current world's best values for these decay modes.