Guitar Pattern Question (Major Thirds Tuning)

Question

     Frets (1-4)
       . . . .
       . . . .
       . . . .
       . . . .

Hey everybody. The four dots above are the four frets (vertical) and four strings (horizontal) of the imaginary guitar (sorry I'm not good at giving visual help). Here I'm wondering how many possible chord shapes I can possibly make within the range of 4 strings and four frets using four (always four) coordinates at a time. Here are the examples I'm looking for

            . . x .               x . . .
            . . x .               . x . .
            x . . .               . . x .
            . x . .               . . . x

The x's represent where I'm fretting on the fingerboard.

Now here are the restrictions (examples I don't want counted)

          . . X X                   x . . .
          . . X .                   . x x .
          . . . X                   . x . .
          . X . .                   . . . .

If there are two or more X's within the same string then it doesn't work due simply to the nature of the instrument. The second example is missing one string fretted so that's a no-no also in my book (I'm looking for all 4 strings to be fretted).

One more restriction!

                        X . . .        . X . .
                        . X . .        X . . .
                        . . . X        . . . X
                        X . . .        . X . .

This also I can't allow to be counted as well since the Bottom and Top String is fretted at the same fret. This is just a specific rule though thanks to the nature of my guitar tuning (just the Bottom and Top String). I don't want to count chord shapes that contain repeating notes.

So there you have it! Hopefully a Math Wizard would find out how many possible 4 note shapes I can form with these restrictions put in place.

Part 2: Major Thirds Tuning Guitar Pattern Question Part 2

Part 3: Guitar Patterns Major Third Tuning Part 3

Answer 1

$4$ possibilities for each of $3$ strings, $4-1=3$ possibilities for the last string (excludes the first string's fret), gives $4^3 \times 3 = 192$ note shapes.

Answer 2

For selecting note in first row, you have 4 possible ways. First, second, third or fourth.

In the second row you have 4 possibilities for selection of note. You can select any note including one which is just beneath the note you selected from first row.

Coming to next row i.e. third, You again 4 ways to select note. Reason is similar as above.

For the last row, you can select note in 3 ways leaving the note which is in the same column as of first row.

So you have a total of $4 \times 4 \times 4 \times 3 =\boxed{192}$ ways.