Question 1116222
With {{{12}}} points, you can form
{{{12*11*10*9/(4*3*2)=495}}} sets of {{{4}}} points.
Given {{{4}}} points, A, B, C, and D, such that no three points are collinear,
there are {{{3}}} different possible ways to connect them with a closed point to point polygonal:
A-B-C-D-A,
A-B-D-C-A, and
A-C-B-D-A.
In some cases, all three polygonals will be recognized by anyone as a quadrilateral:
{{{drawing(300,300,-1.2,1.2,-0.4,2,
circle(-1,0,0.03),circle(1,0,0.03),
circle(0,0.5,0.03),circle(0,1.732,0.03),
locate(-1.1,-0.02,A),locate(1.05,-0.02,B),
locate(0.05,0.6,C),locate(-0.03,1.9,D),
green(line(-1,0,1,0)),green(line(1,0,0,0.5)),
green(line(0,0.5,0,1.732)),green(line(0,1.732,-1,0)),
locate(-0.3,-0.2,A-B-C-D-A)
)}}} , {{{drawing(300,300,-1.2,1.2,-0.4,2,
circle(-1,0,0.03),circle(1,0,0.03),
circle(0,0.5,0.03),circle(0,1.732,0.03),
locate(-1.1,-0.02,A),locate(1.05,-0.02,B),
locate(0.05,0.6,C),locate(-0.03,1.9,D),
red(line(-1,0,1,0)),red(line(1,0,0,1.732)),
red(line(0,0.5,0,1.732)),red(line(0,0.5,-1,0)),
locate(-0.3,-0.2,A-B-D-C-A)
)}}} , {{{drawing(300,300,-1.2,1.2,-0.4,2,
circle(-1,0,0.03),circle(1,0,0.03),
circle(0,0.5,0.03),circle(0,1.732,0.03),
locate(-1.1,-0.02,A),locate(1.05,-0.02,B),
locate(0.05,0.6,C),locate(-0.03,1.9,D),
blue(line(-1,0,0,0.5)),blue(line(1,0,0,0.5)),
blue(line(1,0,0,1.732)),blue(line(0,1.732,-1,0)),
locate(-0.3,-0.2,A-C-B-D-A)
)}}} .
 
In other cases, some of the polygonals may include segments intersecting each other, and may not agree with everyone's idea of a quadrilateral:
{{{drawing(300,300,-1.2,1.2,-0.4,2,
circle(-1,0,0.03),circle(1,0,0.03),
circle(-0.9,1.75,0.03),circle(0.8,1.75,0.03),
locate(-1.1,-0.02,A),locate(1.05,-0.02,B),
locate(0.85,1.9,C),locate(-0.95,1.9,D),
green(line(-1,0,1,0)),green(line(1,0,0.8,1.75)),
green(line(-0.9,1.75,0.8,1.75)),green(line(-1,0,-0.9,1.75)),
locate(-0.3,-0.2,A-B-C-D-A)
)}}} , {{{drawing(300,300,-1.2,1.2,-0.4,2,
circle(-1,0,0.03),circle(1,0,0.03),
circle(-0.9,1.75,0.03),circle(0.8,1.75,0.03),
locate(-1.1,-0.02,A),locate(1.05,-0.02,B),
locate(0.85,1.9,C),locate(-0.95,1.9,D),
red(line(-1,0,1,0)),red(line(1,0,-0.9,1.75)),
red(line(-0.9,1.75,0.8,1.75)),red(line(-1,0,0.8,1.75)),
locate(-0.3,-0.2,A-B-D-C-A)
)}}} , {{{drawing(300,300,-1.2,1.2,-0.4,2,
circle(-1,0,0.03),circle(1,0,0.03),
circle(-0.9,1.75,0.03),circle(0.8,1.75,0.03),
locate(-1.1,-0.02,A),locate(1.05,-0.02,B),
locate(0.85,1.9,C),locate(-0.95,1.9,D),
blue(line(-1,0,0.8,1.75)),blue(line(1,0,0.8,1.75)),
blue(line(1,0,-0.9,1.75)),blue(line(-1,0,-0.9,1.75)),
locate(-0.3,-0.2,A-C-B-D-A)
)}}} .
 
If we can call every one of the polygonals shown above a quadrilateral,
we can form {{{495*3=1485}}} quadrilaterals.
 
If crisscrossing polygonals are not considered quadrilaterals,
I suspect the number of quadrilaterals depends on the placement of the points,
and I do not know how to predict that number.