1. Write the MO ordering

For second-period homonuclear diatomic molecules:

• For $\mathrm{B_2, C_2, N_2}$:
  $\sigma(2s)<\sigma^*(2s)<\pi(2p_x)=\pi(2p_y)<\sigma(2p_z)$
• For $\mathrm{O_2, F_2}$:
  $\sigma(2s)<\sigma^*(2s)<\sigma(2p_z)<\pi(2p_x)=\pi(2p_y)<\pi^*(2p_x)=\pi^*(2p_y)$

We now find the highest occupied molecular orbital (HOMO) in each species and count its unpaired electron(s).

2. $\mathrm{N_2}$

Each N has 7 electrons, so $\mathrm{N_2}$ has $14$ electrons total.

MO configuration: $\sigma(1s)^2\,\sigma^*(1s)^2\,\sigma(2s)^2\,\sigma^*(2s)^2\,\pi(2p_x)^2\,\pi(2p_y)^2\,\sigma(2p_z)^2$

The HOMO is $\sigma(2p_z)$, which contains 2 paired electrons.

So, number of unpaired electrons in HOMO of $\mathrm{N_2}$ is: $0$

3. $\mathrm{N_2^+}$

Removing one electron from $\mathrm{N_2}$ removes it from the HOMO, i.e. from $\sigma(2p_z)$.

So configuration becomes: $\sigma(2p_z)^1$ for the HOMO.

Hence the HOMO has one unpaired electron.

So, number of unpaired electrons in HOMO of $\mathrm{N_2^+}$ is: $1$

4. $\mathrm{O_2}$

Each O has 8 electrons, so $\mathrm{O_2}$ has $16$ electrons total.

For oxygen, MO order is: $\sigma(2s)<\sigma^*(2s)<\sigma(2p_z)<\pi(2p_x)=\pi(2p_y)<\pi^*(2p_x)=\pi^*(2p_y)$

Configuration: $\sigma(1s)^2\,\sigma^*(1s)^2\,\sigma(2s)^2\,\sigma^*(2s)^2\,\sigma(2p_z)^2\,\pi(2p_x)^2\,\pi(2p_y)^2\,\pi^*(2p_x)^1\,\pi^*(2p_y)^1$

The HOMO consists of the degenerate orbitals $\pi^*(2p_x)$ and $\pi^*(2p_y)$, each with one electron.

Thus, in the highest occupied molecular orbital level, there are: $2$ unpaired electrons.

5. $\mathrm{O_2^+}$

Removing one electron from $\mathrm{O_2}$ removes it from the HOMO $\pi^*$ level.

So configuration becomes: $\pi^*(2p)^3$ spread over two degenerate $\pi^*$ orbitals. One orbital has a pair and the other has one single electron.

Thus, number of unpaired electrons in HOMO of $\mathrm{O_2^+}$ is: $1$

6. Final sequence

For $\mathrm{N_2;\ N_2^+;\ O_2;\ O_2^+}$

the number of unpaired electrons in the HOMO are: $0,\ 1,\ 2,\ 1$

This matches Option C.

7. Comparison with stored answer

Stored correct answer: C
Derived answer: C

Hence, the derived answer agrees with the stored correct answer.