Which of the following sets of quantum numbers is not allowed?

The rules for quantum numbers are:

- Principal quantum number: $$n = 1, 2, 3, \ldots$$

- Azimuthal quantum number: $$l = 0, 1, 2, \ldots, (n-1)$$

- Magnetic quantum number: $$m_l = -l, -(l-1), \ldots, 0, \ldots, (l-1), l$$

- Spin quantum number: $$s = +\dfrac{1}{2} \text{ or } -\dfrac{1}{2}$$

Let us check each option:

Option A: $$n = 3, l = 2, m_l = 0, s = +\dfrac{1}{2}$$ — Valid. For $$n = 3$$, $$l$$ can be 0, 1, or 2. For $$l = 2$$, $$m_l$$ can range from $$-2$$ to $$+2$$.

Option B: $$n = 3, l = 2, m_l = -2, s = +\dfrac{1}{2}$$ — Valid. Same reasoning as above.

Option C: $$n = 3, l = 3, m_l = -3, s = -\dfrac{1}{2}$$ — Not allowed. For $$n = 3$$, the maximum value of $$l$$ is $$n - 1 = 2$$. So $$l = 3$$ is not permitted.

Option D: $$n = 3, l = 0, m_l = 0, s = -\dfrac{1}{2}$$ — Valid. For $$l = 0$$, $$m_l$$ must be 0.

Therefore, the correct answer is Option C.