Question

There are 12 cars contesting a race. The first three cars completing the race will be awarded prizes. In how many ways can the prizes be awarded?

Answer 1

If the prizes are unique, 1320 different ways. If the prizes are the same, 220 different ways.

Explanation:

If the prizes are unique (such as First, Second, and Third place prizes), then we have a permutation (because we care about the order in which the cars come in). The permutation general formula is:

`P_(n,k)=(n!)/((n-k)!); n="population", k="picks"`

`P_(12,3)=(12!)/(9!)=12xx11xx10=1320` different ways

If, however, the prizes aren't unique (such as when a race is a qualifier to a bigger race - the first three cars are given the ability to move on while the other 9 are eliminated), then we have a combination (because we don't care about the order in which the cars come in). The combination general formula is:

`C_(n,k)=((n),(k))=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

`C_(12,3)=((12),(3))=(12!)/((3!)(9!))=(12xx11xx10)/6=220` different ways