At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?

$ \textbf{(A)}\ 8\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 18\qquad
\textbf{(E)}\ 24$
1. Let's denote the number of women at the party as \( w \).
2. Each man danced with exactly 3 women. Since there are 12 men, the total number of dances involving men is:
   \[
   12 \times 3 = 36
   \]
3. Each woman danced with exactly 2 men. Therefore, the total number of dances involving women is:
   \[
   2w
   \]
4. Since the total number of dances involving men must equal the total number of dances involving women, we set up the equation:
   \[
   36 = 2w
   \]
5. Solving for \( w \):
   \[
   w = \frac{36}{2} = 18
   \]

The final answer is \( \boxed{18} \).