Rules for allowed quantum numbers

For an electron, the quantum numbers must satisfy:

• Principal quantum number: $n = 1,2,3,\dots$
• Azimuthal quantum number: $l = 0$ to $(n-1)$
• Magnetic quantum number: $m_l = -l$ to $+l$
• Spin quantum number: $s = +\frac{1}{2}$ or $-\frac{1}{2}$

Check each option

Option A

$n=3,\ l=2,\ m_l=0,\ s=+\frac{1}{2}$

• For $n=3$, allowed values of $l$ are $0,1,2$
• Here $l=2$ is allowed
• For $l=2$, $m_l$ can be $-2,-1,0,+1,+2$
• Here $m_l=0$ is allowed
• Spin $s=+\frac{1}{2}$ is allowed

So, A is allowed.

Option B

$n=3,\ l=2,\ m_l=-2,\ s=+\frac{1}{2}$

• For $n=3$, $l=2$ is allowed
• For $l=2$, $m_l=-2$ is allowed
• Spin $s=+\frac{1}{2}$ is allowed

So, B is allowed.

Option C

$n=3,\ l=3,\ m_l=-3,\ s=-\frac{1}{2}$

• For $n=3$, allowed values of $l$ are only $0,1,2$
• But here $l=3$, which is not allowed

Hence, C is not allowed.

Option D

$n=3,\ l=0,\ m_l=0,\ s=-\frac{1}{2}$

• For $n=3$, $l=0$ is allowed
• For $l=0$, the only allowed value of $m_l$ is $0$
• Spin $s=-\frac{1}{2}$ is allowed

So, D is allowed.

Conclusion

The set of quantum numbers that is not allowed is: $\boxed{\text{C}}$

Comparison with stored correct answer

Stored correct answer = C

My derived answer = C

Therefore, they agree.