Small signal equivalent circuit BJT question
Without using any small-signal equivalent diagram I consider the circuit as a two-stage transistor amplifier: (1) The first stage is in common-collector configuration and (2) the 2nd stage is a clasical common-emitter stage.


Assuming a DC voltage of 0.6 volts across both B-E diodes it is easy to find the base current IB1 for the first transistor and, hence, all the other three currents (IC1, IB2, IC2) because the current gains are given.
Applying the relation for the transconductance gm=Ic/VT and rbe=beta/gm we easily can find the gain values for both stages: G1=0.5 and G2=-100. Hence, the total gain, indeed, is G=G1*G2=-50.


(By the way: In a typical/classical Darlington configuration both collector nodes are connected).