If all of the methyl groups were in the equatorial conformation there would be more steric hindrance caused by the closer hydrogens. Axial conformation of one of those groups would reduce the steric hindrance between them.
This isn't quite true. You're talking about gauche butane interactions, but they are not applicable to this discussion. The compound is cis-1,2,4-trimethylcyclohexane. All 3 methyl groups are cis to each other, so they are all on the same face. So we have two possible chair conformations:
first: at the 1 position, we're equatorial, and the two position, we're axial, and at the 4 position we're axial (because if you take a single face of a chair conformation of a cyclohexane, the substituents on that face will alternate between axial and equatorial. so if we pick equatorial at the 1 position, whatever's on that face at the 2 position is axial, at the 3 position it's equatorial, at 4 it's axial, at 5 it's equatorial, and at 6 it's axial)
second: at the 1 position, we're axial, at the two position we're equatorial, and at the 4 position we're equatorial.
these are the only two possibilities we have, because we are limited by the fact that they tell us that the methyl groups are all on the same face of the molecule.
so first of all, that eliminates A as an answer choice because it isn't physically possible to have cis-1,2,4-trimethylcyclohexane exist in an all equatorial conformation since the chair form would place either one or two of those groups axial. Now, if we had the 1,2,4-trimethylcyclohexane where the 2 and 4 were cis to each other and the 1 methyl was trans to the other two methyls, then yes, the preferred conformation is the all-equatorial conformation.
so our answer is either b or c. B, which is the second of the two possibilities I mentioned above, is correct because it places the most number of groups equatorial. Placing two groups axial, as in the first possibility I mentioned, is extremely unfavorable, because there is a Me/Me 1,3-diaxial interaction between the methyl groups at positions 2 and 4. Placing just 1 group axial is better, since the primary steric interaction is the Me/H 1,3-diaxial interaction.
The point sharing was making about having two methyl groups in adjacent equatorial positions is, in fact, an unfavorable steric interaction. However, it cannot apply to this problem because the only way you can have two adjacent methyl groups both in equatorial positions is if they are trans to each other. In addition, if you did have a case where there were two methyl groups in adjacent positions trans to each other, ring flipping to put both of them axial would only introduce more strain on the molecule by creating 1,3-diaxial interactions.
lemme know if you have questions, I'll have to check what I wrote in the morning to make sure it's coherent
