To get the total number of posiblilities you must first relize that some sudokus have the same pattern.
Take
123456789
456789123
789123456
234567891
567891234
891234567
345678912
678912345
912345678
and
637159428
159428637
428637159
371594286
594286371
286371594
715942863
942863715
863715942
for example.
If you take the 1st box and you call the
up-left A
up B
up-right C
left D
center E
right F
bottom-left G
bottom H
bottom-right I
You'll get out of both sudoku's:
ABCDEFGHI
DEFGHIABC
GHIABCDEF
BCDEFGHIA
EFGHIABCD
HIABCDEFG
CDEFGHIAB
FGHIABCDE
IABCDEFGH
Cause you can connect 123456789 to ABCDEFGHI on 362880 (9*8*7*6*5*4*3*2*1 = 9!) ways.
So the formule we can now conlude is:
possiblilities = patterns * 362880.
The number of patterns is based on how many we ways we can place ABCDEFGHI in the sudoku
Now lets see on how many ways we can place the A in the sudoku:
Diffrent colors indicates the boxes (except for the 1st)
White means unusable place.
A black dot indicates the position of A.
A light color indicates where the black dot of that box also could have been placed.
A dark color indicates where the black dot could be placed if the other black dots where diffrently placed.
You might think "What if i place the 'A' in the first block on position 2?" Well if we do that we get on how many ways we can set the 'B'
And now a image for the rest:
The number of posibliities we can place 'A' in a box per box is:
1 6 3
6 4 2
3 2 1
Thats 5184 (1*6*3*6*4*2*3*2*1).
The same as for BCDEFGHI
But, if we do
5184^8 * 362880
It's wrong cause some pattrens overlap eachother.
And i'm currently working on how many pattrens overlap.
What I do know is that the forumla is something like:
5184 * ? * ? * ? * ? * ? * ? * ? * 1 * 362880
And the overlapping calulating has something to do with 2^?.
Thats how far i got.
Update/edit:
I made a table of in how many times the 'A' is set in a place in the box in all of it's patterns: