Step-by-step Solution:

We are given two sets:
- \( A \) contains 4 elements.
- \( B \) contains 2 elements.

Step 1: Find the number of elements in \( A \times B \)
The Cartesian product \( A \times B \) consists of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \).

Since \( A \) has 4 elements and \( B \) has 2 elements, the number of elements in \( A \times B \) is:
\[ |A \times B| = 4 \times 2 = 8 \] Step 2: Find the total number of subsets of \( A \times B \)
A set with \( n \) elements has \( 2^n \) subsets. So, the total number of subsets of \( A \times B \) is:
\[ 2^8 = 256 \] Step 3: Find the number of subsets with at least 3 elements
The subsets of \( A \times B \) include:
- The empty set (1 subset)
- Subsets with 1 element (choosing 1 out of 8: \( \binom{8}{1} = 8 \))
- Subsets with 2 elements (choosing 2 out of 8: \( \binom{8}{2} = 28 \))
Thus, the number of subsets with fewer than 3 elements is: \[ 1 + 8 + 28 = 37 \] Therefore, the number of subsets with at least 3 elements is: \[ 256 - 37 = 219 \] Answer: \(\mathbf{C. \ 219}\)