The LHC is the highest energy particle collider in the world. The LHCb experiment is one of the four main experiments at the LHC. Between 2015 and 2018 it collected 5.4 fb ̄1 of data at a centre of mass energy of 13 TeV. In this thesis, the cross-section for tt ̅ μeb production is measured using this data set. The study of this measurement is a test of the predicting power of Standard Model theory and also contributes to constraining future predictions of proton interactions. A selection scheme is developed to reduce the expected contributions from a number of backgrounds and improve the purity of the final state. The finnal measurement, σtt ̄, represents the cross-section of the production of top quark pairs decaying to a muon, electron and b-jet, where the ρτ of each object is above 20 GeV, the lepton pseudorapidities are between 2 and 4.5, and the b-jet pseudorapidity is between 2.2 and 4.2. The result is: σtt ̄ = 117±10(stat.) ±15(syst.)±5(lumi.)pb where the first uncertainty is statistical, the second is systematic and the third is due to the luminosity determination. Comparisons are made with theoretical predictions calculated from POWHEG and aMC@NLO and the measurement is seen to be in good agreement. A feasibility study of the measurement of the WW production cross-section is also performed. While the measurement will be possible at LHCb in the near future, it is not feasible with the current data set, due mainly to the limited statistics available. It is predicted that a significant measurement of this cross-section will be possible by the end of LHCb Run-III.