This is potentially a community, because there’s no reason to think the soldiers were able to select their battalion, or that the specific battalion is attracting soldiers who self-selected into it.

This is potentially a community, because there’s at least some socioeconomic diversity, and there’s likely to be diversity of career and personal interests among the students. Joining the same campus political organization might suggest self-selection, but this isn’t as clear evidence of self-selection as what we get in (C).

This is least likely to be a community, because the group of doctors is a self-selecting group drawn together by professional interest. Even though you might think some other answers involve groups that probably aren’t communities, either, (C) provides the strongest evidence that the group self-selects based on a shared interest. In addition, there’s nothing in (C) that suggests there’s genuine diversity along any of the dimensions the author mentions. No other answer both provides evidence of self-selection and lacks diversity. That’s why the author would think (C) is the least likely to be a community.

This is potentially a community because there’s no reason to think the teachers self-selected to work in the same school because of some shared interest.

This is potentially a community because there’s at least a high degree of economic and cultural diversity.