Final answer:
The appropriate expression to fill in the blank is 'sin darse cuenta'. It means 'without realizing'. 'De repente', 'después de todo', and 'dentro de un rato' are not suitable choices.
Step-by-step explanation:
The appropriate expression to fill in the blank in the sentence is sin darse cuenta, which means 'without realizing'. This expression indicates that the teacher gave the students the answers to the exam unintentionally or without being aware of what she was doing.
For example, the sentence could be rewritten as 'La profesora les dio las respuestas del examen a los alumnos sin darse cuenta (sin darse cuenta de lo que estaba haciendo)'. This emphasizes the teacher's lack of awareness or realization.
It is important to note that 'de repente' (suddenly), 'después de todo' (after all), and 'dentro de un rato' (in a while) are not suitable expressions to convey the intended meaning in the given sentence.
Learn more about Use of expression 'sin darse cuenta' in Spanish sentence