1. Find the number of elements in $A \times B$

Since set $A$ has $4$ elements and set $B$ has $2$ elements, $|A \times B| = |A|\cdot |B| = 4\cdot 2 = 8.$

So, $A \times B$ is a set with $8$ elements.

2. Total number of subsets of $A \times B$

A set with $8$ elements has $2^8 = 256$ subsets.

3. Count subsets having at least 3 elements

We need subsets with sizes: $3,4,5,6,7,8.$

Instead of adding all of these directly, subtract subsets with fewer than $3$ elements from the total.

Subsets with fewer than $3$ elements are:

• $0$-element subsets: $\binom{8}{0}=1$
• $1$-element subsets: $\binom{8}{1}=8$
• $2$-element subsets: $\binom{8}{2}=28$

Total such subsets: $1+8+28=37.$

Therefore, the number of subsets having at least $3$ elements is $256-37=219.$

4. Check with options

The correct option is: $\boxed{219}$ which is Option A.

5. Compare with stored correct answer

Stored correct answer = A

Our derived answer = A

So, they agree.