For example F major chord = F0, A0, C1.
Would chords like "C0, F0, A0" or "FO, A1, C2" still be F major chords?
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Sign up to join this communityYes. As long as you play (only) the pitches F, A, and C, it's an F chord regardless the order, spacing, doubling, or octaves of the notes.
If the notes are packed as closely together as possible, it's called "close position". If one or more are spaced apart, it's called "open position", which is what you're describing.
If the F is the lowest pitch, it's called "root position." If the A is the lowest pitch, it's called "first inversion". If the C is the lowest pitch, it's called "second inversion."
The F chord comprises three notes - F A and C. It could even be argued that the C isn't necessary - although that would make it a 'two note chord' - which to a lot of folk isn't even a chord then.
Those three notes, then, when played simultaneously, make the triad known as F major. It matters not which octave any of the three notes are in, it's still F major. True, some mixtures will sound really odd, but it doesn't change the fact. You could also have a 100 piece orchestra, with each individual instrument playing one of those notes as chosen, in many different octaves, some playing the same letter name, in different ways, and it's still F major.
Aaron's answer covers inversions and positions well. Your chord voicing puts it into root, close position.