The following word problem was given as a challenge.

I am a Mother of an Honors Algebra 8th Grader....trying to remember how to do these problems.

First count the number of ways if there are no ties. In this case any one of the 3 horses could finish first and, once you know who is first, the other two hourses can finish second and third or third and second. Hence there are 3  2 = 6 different ways the hourses can finish if there are no ties.

Now, how many outcomes are there if exactly 2 horses are tied for second? There are 3. Any one of the 3 horses could finish first and then the other two are tied.

Similarily there are 3 outcomes with exactly 2 horses tied for first.

All that remains is the 1 situation where all three horses tie for first. Thus, in total there are 6 + 3 + 3 + 1 = 13 different outcomes.