$n\left(A\times B\right)=n\left(A\right)\times n\left(B\right)=2\times 4=8$
$\therefore$ total number of subsets of $A\times B$ are $2^8=256$
For the number of subsets not having more than $5$ number of elements.
The required subsets $=$total number of subsets$-$(sets having $6$ elements) $-$ (sets having $7$ elements) $-$ (sets having $8$ elements)
$=256-\frac{8\times 7}{2}-8-1=219$