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

How do I solve the following problem?

*Twelve horses run a race.*

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

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

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### 2 Answers

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.**

Lets Assume that there is a winner. In that case the possibilities are:-

1. Neck and Neck

2. Participant 1 - First and Participant 2 Second etc.

Part (b) with multiple horses (you don't specify a number) the above scenario will apply and there may be a multiple scenario 1. for the first, second, third and subsequent place.