Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawig the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies on between the parallel lines (excluding the intersection points on the lines) ?

$ 
\textbf{(A)}\ 5600
\qquad\textbf{(B)}\ 5650
\qquad\textbf{(C)}\ 6500
\qquad\textbf{(D)}\ 6560
\qquad\textbf{(E)}\ 6600
$
1. **Understanding the Problem:**
   We need to find the maximum number of intersection points of segments connecting black points on line \(d_1\) with white points on line \(d_2\). The intersection points must lie between the parallel lines \(d_1\) and \(d_2\).

2. **Counting the Segments:**
   - There are 11 black points on \(d_1\).
   - There are 16 white points on \(d_2\).
   - Each black point can be connected to each white point, resulting in \(11 \times 16 = 176\) segments.

3. **Intersection of Segments:**
   - Consider two black points \(B_1\) and \(B_2\) on \(d_1\).
   - Consider two white points \(W_1\) and \(W_2\) on \(d_2\).
   - The segments \(B_1W_1\) and \(B_2W_2\) will intersect if and only if the points are chosen such that \(B_1\) and \(B_2\) are on different sides of \(W_1\) and \(W_2\) when projected onto the line connecting the points.

4. **Counting the Intersections:**
   - To count the number of such intersections, we need to choose 2 black points from the 11 black points and 2 white points from the 16 white points.
   - The number of ways to choose 2 black points from 11 is given by the binomial coefficient \(\binom{11}{2}\).
   - The number of ways to choose 2 white points from 16 is given by the binomial coefficient \(\binom{16}{2}\).

5. **Calculating the Binomial Coefficients:**
   \[
   \binom{11}{2} = \frac{11 \times 10}{2 \times 1} = 55
   \]
   \[
   \binom{16}{2} = \frac{16 \times 15}{2 \times 1} = 120
   \]

6. **Total Number of Intersections:**
   - The total number of intersection points is the product of these two binomial coefficients:
   \[
   \binom{11}{2} \cdot \binom{16}{2} = 55 \cdot 120 = 6600
   \]

Conclusion:
The maximum number of points of intersection of the segments that lie between the parallel lines is \(\boxed{6600}\).