An instructor gives her class a set of 10 problems with the information that the final exam will consist of a random selection of 5 of them. If a student has figured out how to do 7 of the problems, what is the probability that he or she will answer correctly

(a) all 5 problems?

So the sample space, is

will be the event that the student doesn't know all 5 problems. that he doesn't know only 4 questions, and so on.

Since the student knows 7 problems the event that the test contains more then 3 unknown problems has to be a null set as there are at most 3 unknown problems.

OK so there are two pools of problems, the seven that he knows and the 3 that he doesn't. So and where and only four valid combinations that sum to five, . So,

So and

This is the listed answer but does this process look good/correct?

Also

(b) at least 4 of the problems?

So this should just be

Lastly the probablility is the quotient of the and

Which is the given answer. Could someone be kind enough to double check this for me?