161 729 840 combinations of pieces around top-left clue piece (3x3 puzzle)

161 729 840 combinations of pieces around upper-left clue piece (3x3 puzzle), with the 4 clue pieces and the mandatory piece fixed (see first illustration below).

If only mandatory piece was fixed (and the top-left clue piece of course), this would give 185 590 785 combinations(see second illustration below).

With the mandatory piece unfixed (yes, this is stupid), this 3x3 partial puzzle would have 192 340 030 combinations (see third illustration below).

For the 3 other clue pieces, with all clue pieces and mandatory piece fixed, there is :

143 807 839 combinations of pieces around top-right clue piece
144 580 904 combinations of pieces around bottom-right clue piece
137 642 043 combinations of pieces around bottom-left clue piece