Solution

Given: \(A=\{1,2,3,4\}\), \(B=\{3,4,5\}\)

\(A\cup B=\{1,2,3,4,5\}\Rightarrow |A\cup B|=5\)
\(A\cap B=\{3,4\}\Rightarrow |A\cap B|=2\)
\(A\triangle B=(A\cup B)\setminus(A\cap B)=\{1,2,5\}\Rightarrow |A\triangle B|=3\)

Size of Cartesian product: \(|(A\cup B)\times(A\cap B)\times(A\triangle B)| = 5\times 2\times 3 = \mathbf{30}\).