Understand the rules for quantum numbers.

Step 1: Understand the rules for quantum numbers. For a set of quantum numbers $(n, l, m, s)$ to be valid:

• The principal quantum number $n$ must be a positive integer ($n \ge 1$).
• The azimuthal quantum number $l$ must be an integer from $0$ to $n-1$.
• The magnetic quantum number $m$ must be an integer from $-l$ to $+l$.
• The spin quantum number $s$ must be either $+\frac{1}{2}$ or $-\frac{1}{2}$.

Step 2: Evaluate each option based on these rules.

• A. $n=2, l=0, m=2, s=-\frac{1}{2}$

  • $n=2$ is valid.
  • $l=0$ is valid (since $0 \le 0 \le n-1=1$).
  • $m=2$ is invalid because for $l=0$, $m$ must be $0$. The value $m=2$ is not allowed.

• B. $n=2, l=0, m=1, s=+\frac{1}{2}$

  • $n=2$ is valid.
  • $l=0$ is valid (since $0 \le 0 \le n-1=1$).
  • $m=1$ is invalid because for $l=0$, $m$ must be $0$. The value $m=1$ is not allowed.

• C. $n=3, l=0, m=3, s=-\frac{1}{2}$

  • $n=3$ is valid.
  • $l=0$ is valid (since $0 \le 0 \le n-1=2$).
  • $m=3$ is invalid because for $l=0$, $m$ must be $0$. The value $m=3$ is not allowed.

• D. $n=2, l=1, m=1, s=+\frac{1}{2}$

  • $n=2$ is valid.
  • $l=1$ is valid (since $0 \le 1 \le n-1=1$).
  • $m=1$ is valid (since for $l=1$, $m$ can be $-1, 0, 1$. The value $m=1$ is allowed).
  • $s=+\frac{1}{2}$ is valid.

Step 3: Conclude which set is possible. Only option D contains a set of quantum numbers that are all valid according to the rules.

The final answer is $\boxed{\text{D}}$.