Nonmusical scales X.
Fingering scales by Sápi III.
This section contains Sápi fingering scales that have a range of three strings.
The three strings give a total number of 81 variations (3^{4}). Combining these with the standard fingering 24 combinations will have a result of 81 x 24 = 1944. However, the variations that only involve one or two strings will be deducted from this number (we have 3 onestring variation, and 3 x 14 = 42 twostring variations), so to get the final number we will use the following formula: 81  (3 + 42) = 36.
36 variations where all three strings are used, combined with the standard 24 fingering combinations, and the result is 36 x 24 = 864.
Let's see what we've got so far:

864 threestring variations

That makes altogether 1224 variations!
I have copied here how Zoli SĂˇpi calculates the number of variations. Although it is a very much different calculation, the end result is still the same.
...The four notes can only be distributed on the three strings in a way where there will always be a string that has two notes on it. To calculate the number of these occasions, we need to use the following formula:
where

n = 4

k = 2
Thus 4! / 2! * (4  2)!) x 3 =18. And since the remaining two fingers on the other two strings can be interchanged, we need to double it: 18 x 2 = 36. This is the number of variations for four fingers placed on three strings, on consecutive frets.
This 36 variations need to be multiplied by the 24 fingering combinations, that is 36 x 24 = 864. In the below charts, the same colors mean same picking technique in a given scale group. All scales are played with alternate picking...
211 scales
2210
2120
2102
1220
1202
1022
2201
2021
2012
0221
0212
0122
121 scales
1120
1210
1201
2110
2101
2011
1102
1012
1021
0112
0121
0211
112 scales
0021
0201
0210
2001
2010
2100
0012
0102
0120
1002
1020
1200
What's the next exercise? The 4string fingering scales!