With the 4 common answers (all 4 people answered them the same) we know that no more than 3 of them can be correct (because Lisa scored 3 points). We also know at least one must be correct (otherwise Mary can't score 7).

Examine what Mary and Dan have in common besides those 4... Nothing. They disagreed on each of the remaining 6 answers.

We can break this down into three cases, either 1, 2, or 3 of the common answers are correct:

1 - Impossible. Mary gets 6 of the remaining correct, Dan gets 4 of the 6 remaining correct, but they have none in common so this doesn't work.

2 - Impossible. Mary gets 5 of the remaining correct, Dan gets 3 of the remaining correct, but again their answers disagree so they cannot share a common answer out of the 6 remaining.

3. Mary needs 4 of the remaining 6 and Dan needs 2, this is feasible.

At this point you can ignore the 4 common ones other than to say Colin has at least a score of 3.

Since we now know that everything outside those common ones are incorrect for Lisa, cross everything off that Lisa and Colin share of the 6 non-common answers. There are only 3 that disagree with Lisa so they must be correct (because everything Lisa answered here is wrong). This gives Colin a final score of 6, three in common with everyone and three from disagreeing with Lisa.

The reason there are 4 keys in Hillel's solution is actually kind of funny, they don't matter. Those 4 keys only disagree on which of the 4 common answers are incorrect. They agree everywhere else.

EDIT: Presentation and some typos