Let $A$ and $B$ be two sets containing $2$ elements and $4$ elements respectively. The number of subsets of $A\times B$ having $3$ or more elements is:

1. $n(A)=2,n(B)=4\Rightarrow n(A\times B)=2\times 4=8$.2. Total subsets of $A\times B$ is $2^8=256$.3. Subsets with $0$ elements: $\left(\genfrac{}{}{0ex}{}{8}{0}\right)=1$.4. Subsets with $1$ element: $\left(\genfrac{}{}{0ex}{}{8}{1}\right)=8$.5. Subsets with $2$ elements: $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=\frac{8\times 7}{2}=28$.6. Subsets with $3$ or more elements = Total - (Subsets with $0,1,2$ elements)$=256-(1+8+28)=256-37=219$.