If $A=\{1,2,3,4\},B=\{3,4,5\}$ then the

number of elements in $(A\cup B)\times (A\cap B)\times \left(A\mathrm{\Delta }B\right)$ is

 

$\begin{array}{l}A\cup B=\{1,2,3,4,5\},n(A\cup B)=5\\ \begin{array}{ll}A\cap B& =\{3,4\},n(A\cap B)=2\\ A\mathrm{\Delta }B& =(A~B)\cup (B~A)=\{1,2\}\cup \left\{5\right\}\\ & =\{1,2,5\}\end{array}\end{array}$

$n\left(A\mathrm{\Delta }B\right)=3$ Hence $\begin{array}{l}n\left(\right(A\cup B)\times (A\cap B)\times (A\mathrm{\Delta }B\left)\right\}\\ =5\times 2\times 3=30\end{array}$