Let use recursive function int rec(int col). Here col means that we have already placed one Queen in every column before col (and store positions somewhere in memory) and now we are iterating rows where we can place Queen in column col. It's very important not to place a Queen at position it will be attacked by other Queens to the left. Also we have to ignore reserved positions.

If recursion reach 9-th column, it mean that we have found new Queen placement and we have to increase some answer counter.

How many recursion iterations will be done?

There is a simpler problem about Rook placement where all squares are free. We can use same method to solve this problem.

It can be noticed that recursion will never visit combinations that lead to some conflicts (when two Rooks thread each other), so number of recursion iterations will be equal to number of combinations of Rook placement (there are 8! ~ 4*10^4 combinations of Rook placement).

It is clear that set of all Queen placements is a subset of Rook placement, so solving your problem will not require more iterations.