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I answered a question just now

Given a set S = {x, y, z}, then all the possible subsets of S are

{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}

where, {} denotes the empty set.

I used all the possible there, and then I realized all of the possible may be better, but I am not so sure.

Similarly, following expressions

expression_1: all the balls in the box

expression_2: all of the balls in the box

Which one (or some other expression) is more idiomatic and clear?

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I have seen both "all the possible subsets" and "all of the possible subsets" used in books on set theory. The meaning is the same. Either may be used.

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When you talk something about every thing or person in a group, you can use "all" or "all of", followed by "the" + plural noun, there's no difference in meaning. So all the phrases "all the possible subsets, all of the possible subsets, all the balls, all of the balls" are grammatical.

However the phrase "all the" is more common than "all of the".

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