If 12 horses run a race in how many ways can the first and second place be won and in how many ways can all the horses finish the race?

Twelve horses run the
race.

First let's find the number of ways in which the
first and the second place can be won. Any of the 12 horses can win the 1st place. Of
the remaining 11 horses any of them can win the 2nd
place.

This gives the number of ways in which in the first
and the second place can be won as 12*11 = 132

Now let's
find the number of ways in which all the horses finish the race. The 1st horse can
finish at any of the 12 places. The 2nd can finish at any of the remaining 11 places.
This goes on till there is one place left for the last horse. The total number of ways
in which all the horses finish the race is 12! =
479001600

The number of ways in which the first and
second place can be won is 132 and the number of ways in which all the horses complete
the race is 479001600