The example in question:
"All the students in our high school like to do volunteer work. Some help at nursing homes every week, the others/others serve as crossing guards from time to time."
There's a problem here: it's possible that either of the choices is correct.
Imagine there are 100 students in your high school. All 100 do volunteer work.
If, for example, 70 help at nursing homes and 30 serve as crossing guards, then the answer is "the others". So we use "the others" if those two voluntary jobs account for all 100 students.
But if, for example, 60 help at nursing homes, 25 serve as crossing guards and 15 do some other type of voluntary work, then the answer is "others". So we'd use "others" if those two voluntary jobs didn't account for all 100 students.
Without any further context, both scenarios are possible.
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My interpretation, if I had to choose, would be that " the others" is incorrect. Here's why:
Strictly speaking, "some" generally means "less than half". If we were talking about more than half, it'd be better to use "most" or "many".
Additionally, "the others" often implies a proportion less than half. If the others in this example (that is, the children who serve as crossing guards) made up the majority, it'd make more sense to say something like: "All do volunteer work. Most serve as crossing guards. Others/the others..."
Because it would make more sense to use a word like "most" or "many" for any proportion greater than half of the students, I would assume that neither the children who help at nursing homes nor the students who serve as crossing guards account for half of the volunteers.
If all students do volunteer work, under 50% help at nursing homes and under 50% serve as crossing guards, then some of the students must not have been mentioned in the paragraph.
If that is true, then "the others" is incorrect and "others" is the right answer.
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That's how I would interpret the situation if I heard this example from a native speaker. Then again, it's possible that they just didn't write the test question very well.