Explanation:

Correct Answer is (e).Sentence I: The students in a vulnerable position since they did not have any other alternative.
Here, the word ‘vulnerable’ has been used in the sense that the students were exposed to the possibility of being attached or harmed, physically or emotionally.
Sentence II: The new home is meant for vulnerable youths and adults who are in need of love and care.
Here, the word ‘vulnerable’ means people with a very weak bent of mind and in need of special attention and care.
Sentence III: Sachin Tendulkar saw India off from that vulnerable position to a thumping victory over the world champion Australia in the Tri-nation tournament.
‘Vulnerable’ refers to weak here. It means India were in a weak position initially from where they cruised to victory thanks to Sachin Tendulkar.
So, all the sentences are correct and that makes option E the correct choice.