a) There are 10 possible winners.
For each of those 10 winners, there are 9 possible runners who could come 2nd.
For each of those 10 winners and 9 second-placers, there are 8 possible runners who can come 3rd.
And so on.
Thus there are different orders that the race can finish in (if you discard the possibility of a dead heat between two finishers).
This is written "10 factorial" and it's about 3.2 million.
b) When you're interested only in gold, silver and bronze you just need the first three, that is .
That'll do for a start, hope it gives you a clue about how to continue.