Im not sure if the Chemists teach you guys what those numbers actually mean and the rules that constrain them, instead of just those fancy orbital s, p, d etc stuff but Ill give you the Physicists explanation (Ive forgotten that Chemistry orbital stuff anyway).
You see, with atomic-orbital systems there are four quantum numbers: n (the principal quantum number), l (the orbital angular momentum number), m (the magnetic number) and s (the spin), but we don't need to worry about spin here since as you might know, it only constrains the way the electrons can be arranged in the orbitals and not the orbitals themselves.
You get those numbers by solving the 3-dimensional Schrodinger equation (the numbers actually specify the mathematical form of the wavefunction but they have their physical interpretations) and when you do that, in the process, it turns out that the number l depends on n: it can only take integer values from 0 to n-1. It also turns out that m depends on l: it can only take integer values from -l to l. Those are your rules for the numbers.
Now we are given the energy level we must stay in 4: so what the question is asking you is to count the number of states [4, l, m]. The only difference is that now your number l can go up to n instead of n-1.
So all you need to do is count the states [4, l, m] with these rules. For example if we have l = 0, we can only have m = 0 so we have the single state [4, 0, 0], if we have l = 1, we can have m = -1, 0, 1 giving us 3 states [4, 1, -1], [4, 1, 0], [4, 1, 1]. Now you need to count for all of them up to l = 4. You should get a total of 25.
Now Im not sure how you were taught this in your Chemistry classes though. If you are taught to memorize those fancy orbital types then unfortunately I suppose youll have to do that but I would hope that they would at least teach you those simple rules.
