A committee with $20$ members votes for the candidates $A,B,C$ by a different election system. Each member writes his ordered prefer list to the ballot (e.g. if he writes $BAC$, he prefers $B$ to $A$ and $C$, and prefers $A$ to $C$). After the ballots are counted, it is recognized that each of the six different permutations of three candidates appears in at least one ballot, and $11$ members prefer $A$ to $B$, $12$ members prefer $C$ to $A$, $14$ members prefer $B$ to $C$. How many members are there such that $B$ is the first choice of them? 

$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{More information is needed}
$
1. Define the number of ballots for each permutation of the candidates \(A, B, C\):
   \[
   \begin{cases}
   p = \#(ABC) \\
   q = \#(ACB) \\
   r = \#(CAB) \\
   s = \#(CBA) \\
   t = \#(BCA) \\
   u = \#(BAC)
   \end{cases}
   \]
   We know that the total number of ballots is 20:
   \[
   p + q + r + s + t + u = 20
   \]

2. Given conditions:
   - 11 members prefer \(A\) to \(B\):
     \[
     p + q + r = 11
     \]
   - 12 members prefer \(C\) to \(A\):
     \[
     r + s + t = 12
     \]
   - 14 members prefer \(B\) to \(C\):
     \[
     p + t + u = 14
     \]

3. Sum the three given conditions:
   \[
   (p + q + r) + (r + s + t) + (p + t + u) = 11 + 12 + 14
   \]
   Simplify:
   \[
   2p + q + 2r + s + 2t + u = 37
   \]

4. Subtract the total number of ballots equation:
   \[
   2p + q + 2r + s + 2t + u - (p + q + r + s + t + u) = 37 - 20
   \]
   Simplify:
   \[
   p + r + t = 17
   \]

5. Use the total number of ballots equation to find the remaining variables:
   \[
   q + s + u = 20 - (p + r + t) = 20 - 17 = 3
   \]

6. Express \(q, s, u\) in terms of \(p, r, t\):
   \[
   q = 11 - (p + r)
   \]
   \[
   s = 12 - (r + t)
   \]
   \[
   u = 14 - (p + t)
   \]

7. Ensure all values are non-negative integers:
   \[
   q = 11 - (p + r) \geq 0 \implies p + r \leq 11
   \]
   \[
   s = 12 - (r + t) \geq 0 \implies r + t \leq 12
   \]
   \[
   u = 14 - (p + t) \geq 0 \implies p + t \leq 14
   \]

8. We need to find the number of members who have \(B\) as their first choice, which is \(t + u\):
   \[
   t + u = t + (14 - p - t) = 14 - p
   \]

9. Since \(p + r + t = 17\) and \(p + r \leq 11\), we have:
   \[
   t = 17 - (p + r) \geq 17 - 11 = 6
   \]
   \[
   t \leq 12 - r
   \]

10. Given \(t \geq 6\) and \(t \leq 12 - r\), we find:
    \[
    t = 7 \text{ or } 8
    \]

11. Therefore, the number of members who have \(B\) as their first choice is:
    \[
    t + u = 7 + 1 = 8
    \]

The final answer is \(\boxed{8}\)