plz check for the mistake in my answer (question is of counting )......i mean plz tel me where n why iam wrong

Question

question- there are 12 points in a plane out of which 5 are collinear. The maximum number of distinct quadrilaterals which can be formed with vertices at these points is_____ ??
my answer-
quadrilateral can be formed of four points, so i take 4 points out of 12 in 12_C_4 ways.
then, these also include the quadrilateral formed by 5 collinear points (which actually form no quadrilaterals) , therefore, i subtract 5_C_4 from 12_C_4 and i get the answer as 490.
BUT THE ANSWER GIVEN IS 420.
plz help me........im totally confused to why im getting the wrong answer....

Answer 1

There are 5 collinear points and 7 other points.

Quadrilaterals can be formed with

2 of the collinear points and 2 other points
1 of the collinear points and 3 other points
0 of the collinear points and 4 other points

The number of quadrilaterals that can be formed is:

(5C2)(7C2) + (5C1)(7C3) + (7C4)
= 10*21 + 5*35 + 35 = 210 + 175 + 35 = 420

Answer 2

Sorry I don't have the time to look into this properly, but is it the case that if you form a quadrilateral using 3 of the 5 co-linear points and one of the other 7, you still don't get a "proper" quadrilateral? So you have to remove these from your set too - 5_C_3 * 7 = 70...