You've identified what you need to do in the title (approximations of distributions). Where are you stuck?
In a multiple choice test have 200 questions, each with four possible answers, of which only one is correct. What is the probability that a student hit between 25 and 30 of 80 questions among 200 of which he knows nothing?
You have tried to do: P(25 < X < 30) = P(X<30) - P(X>25)
it should be
I assume the question actually wants which can be worked out as follows
We will say this is approximated by
.9967-.877361 = .119286
I assume the difference between me and the textbook is due to rounding.