So here the chain length is only 2 carbons long. We visualize a #C-C# bond (note that carbon has 4 valence electrons, and here we have used one on EACH carbon to form the first bond); now each carbon has 3 valence electrons; there are 3 hydrogens (with 1 electron each) to bind to each carbon to give #H_3C-CH_3#, our representation of ethane.
The points to remember? That carbon has 4 valence electrons. Of course for #C#, #Z=6#, i.e. 6 positively charged nuclear particles. There are 2 inner core electrons associated with the carbon atom to give a neutral species, but these are inner core and not involved in bonding. Carbon here is correctly identified as a neutral particle.
