다 떨어지다

fail an exam; come apart

"-다(가)" links one action or state to the next, with 떨어지다, which is both "fall" and "fail to get in". The failing sense is plain here: applied to the Hanyang theatre and film department and didn't get in, "I'll try it and I'll flunk". The physical sense shows up as the alternating pair 붙었다 떨어졌다, stuck together then apart — the tongue and the roof of the mouth, things clumping and separating. Two more lines use it for a price dropping to a level and for a coin lying dropped in the road. The verb in front tells you which reading you're getting.