So root, third and fifth I come up with c, e and g to make c chord major but when I play c chord I see my fingers on c, e and c, muting low e with my thumb to play five strings. I cannot figure out how to make sense of this when I starting reading theory. Can someone explain why the difference?
Yes, C, E and G - correct.
First up - your thumb's job is to stop the neck going backwards when you put your fingers on the fretboard so get that back in its proper position.
Your left hand frets a C-chord - low to high - as 032010
Those six notes are E, C, E, G, C, E - your root, third and fifth in various quantities.
Now, play five strings - missing the sixth. This makes a Root Position chord - the fifth string "C" is the lowest sound played. If you play all six strings you get a first inversion (the third (E) in the bass) but it's still a C-chord. If you fret the chord as 332010 then you get a second inversion (the fifth (G) in the bass) which I reckon sounds quite decent.
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