The exact way to solve this problem is using the binomial distribution, assuming that our random variable of interest is "number of students living in apartments" represented by X and
And we want this probability:
So we see that we satisfy the conditions and then we can apply the approximation.
If we appply the approximation the new mean and standard deviation are:
And then
And we are interested on the following probability:
What you do here is subtract g from f: x^2 - 5x + 8 - (x^2 -4); x^2 - 5x + 8 - x^2 + 4; Rearranging gives you x^2 - x^2 - 5x + 8 + 4 which is, simplified, -5x + 12