$C_2$ has 12 electrons.

Moleculer orbital configuration of $C_2$

= ${\sigma _{1{s^2}}}\,\sigma _{1{s^2}}^ * \,{\sigma _{2{s^2}}}\,\sigma _{2{s^2}}^ * \,{\pi _{2p_x^2}}\, = \,{\pi _{2p_y^2}}$

Here no unpaired electron present, so it is diamagnetic.

Here, Bond order = ${{8 - 4} \over 2}$ = 2. So here two bonds are present. For this two bond formation 4 electrons are required which is present in ${\pi _{2p_x^2}}\,$ and $\,{\pi _{2p_y^2}}$

So, in C₂ molecule, only $\pi$ electron occupied the bonding molecular orbital.