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The number of Sudoku combinations

Combination of numbers

How many combinations are there in Sudoku? I thought I could solve this using math I learned in high school, but it’s quite difficult.
The first column is easy, 9*8*7*6*5*4*3*2*1 yields 362880 combinations.
The second column ABC excludes ABC from the first column, so 6*5*4 yields 120.
The third column ABC is 3*2*1, which yields 6.
Up until here, it’s pretty easy, but from the DEF on the second column, we must exclude the DEF on the first column and ABC in the second column.
If the DEF and ABC in the first and second column respectively do not overlap, it is 3*2*1, but it does, so I need to use the opposite combination. But I forgot how to do this, so I’m not sure.
On the 2nd BOX it gets even harder. We have to consider all the numbers in the top row, side row, and the boxes.

It was too hard and I couldn’t solve it, so I looked it up. It seems like there are 6,670,903,752,021,072,936,960 combinations.
This includes symmetrical ones with the same numbers, so if we exclude ones that are rotated 90, 180, or 270 degrees, we get 5,472,730,538 combinations. This figure is based on claims by Mr. Ed Russell and Mr. Frazer.

Jarvis, Frazer; Ed Russell (September 7th 2005). “There are 5472730538 essentially different Sudoku grids ... and the Sudoku symmetry group”. Frazer Jarvis's home page. Posted September 16th 2006