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: