The correct answer is $\alpha {{\text{I}}_{\text{E}}}$.
The equation ${{\text{I}}{\text{C}}} = \beta \,{{\text{I}}{\text{B}}}\, + \,\alpha {{\text{I}}{\text{E}}}$ is the BJT (Bipolar Junction Transistor) model equation. It states that the collector current (${{\text{I}}{\text{C}}}$) is equal to the base current (${{\text{I}}{\text{B}}}$) multiplied by the current gain ($\beta$) plus the emitter current (${{\text{I}}{\text{E}}}$) multiplied by the Early voltage ($\alpha$).
The Early voltage is a parameter that describes the non-linear relationship between the collector current and the base current. It is typically in the range of 100 to 1000 mV.
The current gain ($\beta$) is a parameter that describes how much the collector current is amplified by the transistor. It is typically in the range of 10 to 1000.
The emitter current (${{\text{I}}{\text{E}}}$) is the current that flows into the emitter of the transistor. It is equal to the sum of the collector current (${{\text{I}}{\text{C}}}$) and the base current (${{\text{I}}_{\text{B}}}$).
The BJT model equation is used to calculate the collector current of a transistor. It is a useful tool for designing and analyzing transistor circuits.
The other options are incorrect because they do not represent the correct equation for the collector current of a BJT.