The textbook states that (a) has 2 gauche interactions, (b) has 3 gauche interactions and (c) has 3 interactions.
But how do we determine this? I thought a gauche interaction would occur when two methyl groups are side by side, however I'm unsure where these answers are coming from.
When we sight down a carbon-carbon single bond and draw a Newman projection of a particular conformation, any two groups on the adjacent (vicinal) carbons that have a dihedral angle of 60° between them are said to have a gauche relationship.
In the following figure, the $\ce{X}$ and $\ce{Y}$ have a gauche relationship.
$\ce{X}$ and $\ce{Y}$ can be the same (they can both be methyl groups, other alkyl groups, other atoms such as halogen, etc.) or they can be different from each other. In most organic classes, the gauche interaction between methyl groups or alkyl groups is frequently discussed.
The answers you included (2,3,3) are correct. Looking at the middle molecule (b) in your figure, there are 3 gauche interactions. They are between
