I don't know the picture and I am assuming 10 eyelets. You will need to adjust if any of my assumtions are wrong.
With ten eyelets the lace starts at any of the ten, and in addition it can start on the top side or the underneath of any of the ten eyelets.
Possible 20 start points
There are then 9 eyelets left and it can go through the top or underneath. Possible 18 positions
There are then 8 eyelets left and it can go through the top or the underneath of any of these. Possible 16 positions.
and so on
Answer probably = 20 x 18 x 16 x 14 x 12 x 10 x 8 x 6 x 4 x 2
However if the lace has to go from left to right side all the time (or right to left) then the answer would be
20 x 10 x 8 x 8 x 6 x 6 x 4 x 4 x 2 x 2
So it really depends what instructions you have