For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exacly $2004$ of other lines?

$ 
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 1
$
1. Let \( n \) be the number of lines in the plane, and each line intersects exactly \( 2004 \) other lines. This means each line does not intersect \( n - 2004 - 1 \) lines (since it intersects \( 2004 \) lines out of the total \( n-1 \) other lines).

2. Consider the lines grouped into maximal sets of parallel lines. Each line in a set of parallel lines does not intersect any other line in the same set but intersects all lines in other sets. Let \( k \) be the number of such sets, and let \( a_i \) be the number of lines in the \( i \)-th set. Then, we have:
   \[
   \sum_{i=1}^k a_i = n
   \]
   and each line in the \( i \)-th set intersects \( n - a_i \) lines.

3. Given that each line intersects exactly \( 2004 \) lines, we have:
   \[
   n - a_i = 2004 \quad \text{for each } i
   \]
   which implies:
   \[
   a_i = n - 2004
   \]

4. Since \( a_i \) must be a positive integer, \( n - 2004 \) must be a positive integer, and thus:
   \[
   n > 2004
   \]

5. Additionally, \( n - 2004 \) must be a divisor of \( n \). Let \( d = n - 2004 \). Then:
   \[
   n = 2004 + d
   \]
   and \( d \) must be a divisor of \( 2004 \).

6. The divisors of \( 2004 \) are \( 1, 2, 3, 4, 6, 12, 167, 334, 501, 668, 1002, 2004 \). Each of these divisors corresponds to a possible value of \( d \), and thus a possible value of \( n \):
   \[
   n = 2004 + d
   \]

7. Therefore, the number of different values of \( n \) is equal to the number of divisors of \( 2004 \), which is \( 12 \).

The final answer is \( \boxed{12} \)