1. Find the size of $A \times B$

Given:

• $|A| = 2$
• $|B| = 4$

So, $|A \times B| = |A|\cdot |B| = 2\cdot 4 = 8$

Thus, $A \times B$ has $8$ elements.

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

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

3. We need subsets having 3 or more elements

So subtract subsets having $0,1,$ or $2$ elements.

• Number of subsets with $0$ elements: $\binom{8}{0} = 1$

• Number of subsets with $1$ element: $\binom{8}{1} = 8$

• Number of subsets with $2$ elements: $\binom{8}{2} = 28$

Total subsets with fewer than $3$ elements: $1+8+28 = 37$

4. Required number

$256 - 37 = 219$

So, the number of subsets of $A \times B$ having $3$ or more elements is: $\boxed{219}$

5. Check options

• A: $219$ ✅
• B: $211$ ❌
• C: $256$ ❌
• D: $220$ ❌

Therefore, the correct option is A.