How do I solve the following problem?Twelve horses run a race. (a) How many ways can first and second place be won?...

There are 12 horses running the race. I assume each horse
finishes the race independently and does not share the place with another
horse.

(a) How many ways can first and second
place be won?

The first place can be won by
either of the 12 horses, this gives 12 ways for it to happen. Once the first place has
been secured, the second place can be won by either of the remaining 11 horses. This can
happen in 11 ways. In total we have 12*11 = 132 ways

(b) How many
ways will all the horses finish the race?

There
are 12 horses that can fill the 1st place, 11 that can fill the 2nd and so on till 1
horse can fill the last place. The total number of ways in which all the horses can
finish the race is 12!

The first and second
place can be won in 132 ways. All the horses can finish the race in 12!
ways.