There are 6 candidates. If two refuse to be positioned next to each other, they can be arranged in 480 ways? True or false.  
Six candidates for mayor are to participate in a debate.  Candidates are lined up on stage behind podiums facing the audience.  If two of the candidates refuse to be positioned next to each other, the candidates can be arranged in 480 different ways.
I would normally approach this by simply doing 6! which gives me 720, hence proving the statement false.  The part that throws me is if two candidates are not positioned next to each other.  How can I factor that variable into my formula?
 A: Let $a$ and $b$ be the candidates which do not wish to be positioned next to each other.
You are correct that $6!$ is the number of ways of arranging the $6$ candidates in a line, but this also includes the arrangements where the two unfriendly candidates are positioned next to each other. We thus need to subtract from this the number of ways of arranging the candidates so that $a$ and $b$ are next to each other.
We can count this number that we must subtract in the following way:
We first arrange all of the candidates except $b$ in a line. This can be done in $5!$ ways. We now add $b$ to the line. Since we now want to count the arrangements where $b$ is positioned next to $a$, there are two possibilities for the position of $b$. (Either left of $a$ or right of $a$)
There are thus $2\cdot 5!$ ways of arranging the candidates so that $a$ and $b$ are next to each other, and hence $6! - 2\cdot 5! = 4\cdot 5! = 480$ ways of arranging the candidates so that $a$ and $b$ are not next to each other.
A: It's true. Let those two people be $A$ and $B$. There are totally $6!$ ways to permute people, including the cases that $A$ and $B$ are standing next to each other. To compute the # of cases $A$ and $B$ are together, consider them as a whole, i.e., regard them as a single person since they are standing together. Then there will be $2 \cdot 5! = 240$ cases. $2$ here is to indicate the cases $A$ stands in the left of $B$ and $A$ stands in the right of $B$.
Finally, we have $6! - 240 = 480$.
