I notice that $n^2-3n+2$ can be factored as $(n-2)(n-1)$. So, unless one of the factors is $1$ and the other is prime, the expression will not be prime. That means I only need to check the cases where $n-2=1$ or $n-1=1$. If $n-2=1$, then $n=3$ and $n^2-3n+2=2$, which is prime. If $n-1=1$, then $n=2$ and $n^2-3n+2=0$, which is not prime. So, there is only one positive integer $n$ that makes $n^2-3n+2$ prime, and that is $n=3$. 
For how many positive integers $n$ is $n^2-3n+2$ a prime number?