There are $25$ points on a plane of which $7$ are collinear .
How many quadrilaterals can be formed from these points ?
I did this $^{25}C_{4}-^{7}C_{4}=12615$ quadrilaterals.
But the book is giving answer $^{25}C_{4}-^{7}C_{4}-^{7}C_{3}\times ^{18}C_{1}=11985$ quadrilaterals..
I don't know what is the correct idea .
I look for a short and simple way.
I have studied maths up to $12$th grade.
Best Answer
Your idea is very close to correct. A quadrilateral is formed by $4$ points, where at most $2$ may be colinear. Thus, we have $$ \underbrace{^{25} C_4}_{\text{ choose four points}} - \underbrace{^{7} C_4}_{\text{ subtract out ways to pick four colinear points}} - \underbrace{^{7} C_3 \cdot ^{18} C_1 }_{\text{ subtract out ways to pick three colinear points}}$$