In triangle $ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:

${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 3:4 \qquad\textbf{(C)}\ 4:3 \qquad\textbf{(D)}\ 3:1 }\qquad\textbf{(E)}\ 7:1 } $
1. **Given Information and Assumptions:**
   - In triangle \(ABC\), the ratio \(AC:CB\) is \(3:4\).
   - The bisector of the exterior angle at \(C\) intersects \(BA\) extended at \(P\) (with \(A\) between \(P\) and \(B\)).
   - We need to find the ratio \(PA:AB\).

2. **Angle Definitions:**
   - Let \(\angle CBA = b\) and \(\angle PCA = a\).
   - The exterior angle at \(C\) is \(180^\circ - \angle BCA\).
   - Since the exterior angle bisector divides the angle into two equal parts, we have \(\angle BCA = 180^\circ - 2a\).

3. **Using the Law of Sines:**
   - Assume \(BC = 4\) and \(AC = 3\) (scaling does not change the ratio \(PA:AB\)).
   - By the Law of Sines in \(\triangle ABC\):
     \[
     \frac{BC}{\sin(2a - b)} = \frac{AC}{\sin(b)}
     \]
     Substituting the given lengths:
     \[
     \frac{4}{\sin(2a - b)} = \frac{3}{\sin(b)}
     \]

4. **Finding \(BA\) using the Law of Sines:**
   - By the Law of Sines in \(\triangle ABC\):
     \[
     \frac{BA}{\sin(2a)} = \frac{3}{\sin(b)}
     \]
     Solving for \(BA\):
     \[
     BA = \frac{3 \sin(2a)}{\sin(b)}
     \]

5. **Finding \(AP\) using the Law of Sines:**
   - By the Law of Sines in \(\triangle ACP\):
     \[
     \frac{AP}{\sin(a)} = \frac{3}{\sin(a - b)}
     \]
     Solving for \(AP\):
     \[
     AP = \frac{3 \sin(a)}{\sin(a - b)}
     \]

6. **Finding the Ratio \( \frac{AB}{PA} \):**
   - Using the expressions for \(BA\) and \(AP\):
     \[
     \frac{AB}{PA} = \frac{\frac{3 \sin(2a)}{\sin(b)}}{\frac{3 \sin(a)}{\sin(a - b)}} = \frac{3 \sin(2a) \sin(a - b)}{3 \sin(a) \sin(b)}
     \]
     Simplifying:
     \[
     \frac{AB}{PA} = \frac{2 \cos(a) \sin(a - b)}{\sin(b)} = \frac{\sin(2a - b) - \sin(b)}{\sin(b)}
     \]
     Using the identity \(\sin(2a - b) = \sin(2a - b)\):
     \[
     \frac{AB}{PA} = \frac{\sin(2a - b) - \sin(b)}{\sin(b)} = \frac{4}{3} - 1 = \frac{1}{3}
     \]

7. **Conclusion:**
   - Therefore, the ratio \(PA:AB\) is:
     \[
     PA:AB = \boxed{3:1}
     \]