What is the minimum number of states required to construct a minimal DFA for the language $L = \{w \in \{a, b\}^* \mid w \text{ contains exactly 2 } a\text{'s and at least 2 } b\text{'s}\}$?

To track the count of $a$'s: we need 4 conditions (0, 1, 2, and $>2$ which is a dead/trap state for $a$).

To track the count of $b$'s: we need 3 conditions (0, 1, and $\ge 2$).

Since the occurrences of $a$ and $b$ are independent, the DFA state space is the Cartesian product of the state sets: $4 \times 3 = 12$ states.

Option Analysis: