Solved 5000+ questions on Brilliant!

We can form a quadrilateral in either of the following ways:

i) Selecting 4 points out of 6 non-collinear points = $^6C_4 = ^{6}C_2$

ii) Selecting any 2 points out of 4 collinear points and any 2 out of 6 non-collinear points = $^6C_2 . ^4C_2$

iii) Selecting one point out of the 4 collinear points and any 3 out of 6 non-collinear points = $^6C_3 . ^4C_1$

Note: We can't choose 3 collinear points and 1 which is not collinear with them as then a quadrilateral would not be formed.

Hence, total no. of quadrilaterals = $^6C_2 + ^6C_2 . ^4C_2+ 4. ^6C_3 = \boxed{185}$

Quadrilateral is a four sided-figure.

Number of selection of 4 points out of 10 non-collinear points = $^{10}C_4 = 210$ .

Number of selection of 4 points when no quadrilateral is formed = $^4C_3 \cdot ^6C_1 + ^4C_4 \cdot ^6C_0 = 24 + 1 = 25$

Required Number = $210 - 25 = \boxed{185}$

---

Note : $^nC_r = \dbinom{n}{r}$