Given $n\left(A\right)=4$

$n\left(B\right)=2$

Number of elements in $n\left(A \times B\right)=2\times 4=8$

Number of subsets$=2^8$

Number of subsets having less than $3$ elements ${=}^8{C}_0+{{}^{8}C}_1+{{}^{8}C}_2$

Number of subsets having at least $3$ elements
$=2^8{-}^8{C}_0-{{}^{8}C}_1-{{}^{8}C}_2$

$=256-1-8-28$

$=219$