When I work with a new student who is learning guitar from scratch we first play 4 string versions of G, C and D because they are a bit easier to play. The fact is though, that these chords where you don't use all the strings may actually sound better than "full" chords in some cases.
For example a 4 or even 3 string version of D lets you highlight the higher strings. And an A (or even more commonly, a B chord) played on strings 4, 3 and 2 - where you don't play string 1, can sound perfectly adequate when used the right way.
Rock guitarists actually use two string combinations a lot. So-called "power chords" are 2 string chords played on the lower strings like the well known riff from "Smoke on the Water". And "double stops" are two string combinations played on the higher strings (a la Chuck Berry).
The bottom line is that a "chord" is not always played by strumming across all six strings. To get the best sounding string combinations you need to strike the most appropriate strings.