PX284 - C1b - combining systems

in thermodynamics, considering two systems with entropies, $S_{1}$ and $S_{2}$ , the total entropy is:

S_{t o t} = S_{1} + S_{2}

\begin{aligned} Ω_{t o t} & = Ω_{1} \times Ω_{2} \\ ⟹ S_{t o t} & = k_{B} \ln (Ω_{1} \times Ω_{2}) \\ = k_{B} \ln Ω_{1} + k_{B} \ln Ω_{2} \\ ∴ S_{t o t} & = S_{1} + S_{2} \end{aligned}