Let the number of elements in sets A and B be five and two respectively. Then the number of subsets of A…

Given,

The number of elements in sets $A$ and $B$ be five and two respectively,

So, $n\left(A\right)=5$ and $n\left(B\right)=2$

Now Cartesian product will be, $n\left(A\times B\right)=10$

Number of subsets having three elements ${=}^{10}{C}_3$

Number of subsets having four elements${=}^{10}{C}_4$

Number of subsets having five elements ${=}^{10}{C}_5$

Number of subsets having six elements ${=}^{10}{C}_6$

So, number of subsets having atleast $3$ and at most $6$ elements will be ${=}^{10}{C}_3{+}^{10}{C}_4{+}^{10}{C}_5{+}^{10}{C}_6$

$=120+210+252+210$

$=792$