For a, I think its just 7!. But I don't understand what is the difference between b and c.
I agree with you on the answer to part (a).
With respect to parts (b) and (c), the difference, as I understand it, is between choosing the first three runners in order
("A places first, B places second, and C places third" being distinct from "B places first, A places second, and C places third") and, on the other hand, just naming the top three finishers (where "A, B, and C" is the same as "B, A, and C"). So the difference is between choosing three in order
(that is, using permutation) and without order
(that is, using combinations).
Does that help?