Each group of 3 lines makes one triangle; 
and vise versa, each such a triangle defines a group of 3 lines by an unique way.
So, there are   triangles.


Each group of 4 lines makes one quadrilateral; 
and vise versa, each such a quadrilateral defines a group of 4 lines by an unique way.
So, there are   quadrilaterals.


Each group of 5 lines makes one pentagon; 
and vise versa, each such a pentagon defines a group of 5 lines by an unique way.
So, there are   pentagons.


   . . . . . . . .  and  so  on  . . . . . . 


Each group of 16 lines makes one 16-gon; 
and vise versa, each such a 16-gon defines a group of 16 lines by an unique way.
So, there are   16-gons.


Finally, all 17 lines together make 1 =   17-gon.


So, the number of all closed regions is the sum


    S =  +  +  + . . .  + .


If you complement this sum with the terms   +  + C, you will get


    S +  +  + C =  +  + C +   +  +  + . . .  + .


The long sum in the very right side is equal to  .    (*)


Therefore,  the number S under the question is equal to

    S =  - ( +  + C) =  - 1 - 17 -  =  - 1 - 17 - 136 =  - 154.