# How many election outcomes in the race for class president are there if there are five candidates and 40 students in the class and exactly three candidates tie for the most votes?

##### 1 Answer

10 outcomes if the number of votes doesn't matter. 50 outcomes if we care about the number of votes received by the winners. 560 outcomes if we also care about the number of votes received by the the other two candidates.

#### Explanation:

We have 40 students, five candidates, and the top three tie for the most votes. How many different election results can we achieve?

**If we only care about the number of ways we can see who won and not about the vote count**, we can see that there is a pool of five, three of whom will win. So that gives us:

where

**If we do care about the vote count, read on...**

We know that there are 40 votes cast and there is a three-way tie for the most votes. I'm going to call the vote count for any one of the top three candidates

We have to multiply M by 3, and so we'll have the vote count be:

**Let's first work with #M#**

How large can

How small can

#x, y !=8# because that would then be a 5-way tie- which means either
#x# or#y# would have to be greater than 8, which would mean the 3 candidates who got 8 votes wouldn't have a tie for the most votes.

And so

Ok - we know that

**If we don't care about the details of the results for the two losing candidates**, we can see that for any value of

where

There are five different values for

~~~~~

**What if we do care about the votes received by the losing candidates?**

**How many votes can #x,y# have?**

There are a few limitations:

- We know that
#x,y < M# #x,y >=0# #x+y=40-3M#

Let's try a table to summarize allowable values of

**Now we know how many ways the votes can be cast. Now we need to put candidates to the vote count.**

We know that

There are a total of