Start from case (i)$\Rightarrow$A goes
$\Rightarrow$B goes
$\Rightarrow$Both A And B goes
Let A goes,If A goes $\Rightarrow$C also goes(case(iii)) $\Rightarrow$E wil not go as C is already going((case(ii))$\Rightarrow$No conclusion from (case(iv) and (case(v)) $\Rightarrow$it gives us AC
Let B goes,If B goes then $\Rightarrow$C also goes(case(iii))$\Rightarrow$ AC must goes(case(v)) $\Rightarrow$E will not go as C is already going((case(ii))$\Rightarrow$No conclusion from (case(iv)$\Rightarrow$(case(iii) only tells that AC will be going $\Rightarrow$it gives us AC
Let Both A and B goes$\Rightarrow$AB is going $\Rightarrow$C also goes(case(iii)) $$AC must goes(case(v)) $\Rightarrow$E will not go as C is already going((case(ii))$\Rightarrow$No conclusion from (case(iv)$\Rightarrow$(case(iii) only tells that AC will be going $\Rightarrow$it gives us AC $\Rightarrow$Cthus all total ABC is going.
Our equation simplifies to $AC+AC+ABC$=$AC+ABC=AC\left ( 1+AB \right )=AC$