Take a C major triad. The formula is 1 - 3 - 5, and its notes are C - E - G.

Question 1:

How many inversions are there? I think the answer is 3.
If C is the root note, then we have C - E - G
If E is the root note, E - G - C
If G is the root note, G - C - E

But why C - G - E, E - C - G, G - E - C are not considered as inversions too?

Question 2:

How to play them?
I can play the first inversion C - E - G, because I know the formula (1 - 3 - 5), and I also know how
to play a third and a fifth everywhere on the neck. Alternatively, I can also use the C major scale,
or the C major arpeggio that I can play everywhere. In any case, I can find the "shape" very quickly.

On the other hand, i can't find the other two inversions so easily. Should I rely on the absolute notes,
e.g. I know that the notes are E - G - C, and I need to find them on the neck. Or should I find out what is the formula for the inversion and use that to find how to play it?

I don't really like the first solution: I know the notes on the fretboard, but it means I need to find first what are the notes of the triad. For that, either I use the cycle of fifth (which requires some thinking), or I can visualize the notes on the fretboard. In both cases, it takes some time.

What's the right way to find those shapes?

Thanks,
Philippe

There is only 3 notes in a triad right, so there is only three notes that can can be the bass note. C, E, and G. After you have established the bass note it doesn't matter how you voice the rest of the notes.

Do you know the 3 triad arpeggio shapes? Make them, move them up an octave, and there you have the 3 voicings. Or the way I use to find inversions is start on the bass note, say E, then find the the rest that are in a relatively close location. Say um, E on the 7th fret A string, G on the 5th fret D string, C on the 5th fret G string, and another E on the 5th fret B string. So you make a bar on the 5th fret and just play the E down there on the 7.

Thanks, it's more clear now.