They don't call it the trolley problem for nothing.
If you send the trolley down path (A) to save the people
in path (B) are you responsible for killing the people in (A)?

How can one decide or not decide knowing that doing either
will condemn one group to death while sparing the other group?
Notice I left out murder in exchange for killing because we common
folks tend to conflate the meaning of the two. Killing is more of an
amoral term, so we don't have to address the intention of the act.

Moral absolutism doesn't do so well when pitted with this one.
Here is another interesting one for you, it's about meta-ethics.
There are more than relativism vs. absolutism, and probably a
slew of sub-sects to consider as well, he covers a lot in ten mins.

I think the professor in the Harvard lecture does his best to address the
different schools of thought and how they may see the Trolley problem.