The correct answer is $\boxed{\text{B. }10.1\text{ mA}}$.
The base current is given by $I_B = \dfrac{I_C}{\beta}$, where $\beta$ is the current gain of the transistor. In this case, $\beta = 100$ and $I_C = 10\text{ mA}$, so $I_B = \dfrac{10\text{ mA}}{100} = 0.1\text{ mA}$.
The emitter current is given by $I_E = I_B + I_C = 0.1\text{ mA} + 10\text{ mA} = 10.1\text{ mA}$.
Option A is incorrect because it is the collector current, not the base current. Option C is incorrect because it is the collector current plus the base current, not the emitter current. Option D is incorrect because it is not one of the possible answers.