Answer
See answer below.
Work Step by Step
When the values for $a,b,A$ are given then by the Law of sines: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}$, $\sin B$ can be found.
1) There will exist no triangles, when the sine of the angle $B$ has a value such that the $\sin$ rule is not followed. This means that $B \geq 180^{\circ}-A$
2) There will exist One Triangle when $B=90^{\circ}$.
3) There will exist two triangles, when sine of the angle $B$ is the same for two different values of $c$.
