Let the number of elements in sets A and B be five and two...

Given,

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

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

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 ${=}^{10}{C}_{3}{+}^{10}{C}_{4}{+}^{10}{C}_{5}{+}^{10}{C}_{6}$

$= 120 + 210 + 252 + 210$

$= 792$