The synthesis of t-butyl ethyl ether from sodium ethoxide (C$_2$H$_5$ONa) and t-butyl chloride (C$_4$H$_9$Cl) is achievable via nucleophilic substitution. The strong nucleophile, sodium ethoxide, displaces the chlorine atom in t-butyl chloride, yielding t-butyl ethyl ether. The reaction is represented as: \[ \text{C}_4\text{H}_9\text{Cl} + \text{C}_2\text{H}_5\text{ONa} \rightarrow \text{C}_4\text{H}_9\text{OC}_2\text{H}_5 + \text{NaCl} \] Consequently, t-butyl chloride and sodium ethoxide are the suitable precursors for t-butyl ethyl ether production.