JEE Main & Advanced Mathematics Permutations and Combinations Some Important Results For Geometrical Problems

(1) Number of total different straight lines formed by joining the \[n\] points on a plane of which  \[m(<n)\] are collinear is \[^{n}{{C}_{2}}{{-}^{m}}{{C}_{2}}+1\].

(2) Number of total triangles formed by joining the \[n\]  points on a plane of which \[m(<n)\] are collinear is \[^{n}{{C}_{3}}{{-}^{m}}{{C}_{3}}\].

(3) Number of diagonals in a polygon of \[n\] sides is \[^{n}{{C}_{2}}-n\].

(4) If \[m\] parallel lines in a plane are intersected by a family of other \[n\] parallel lines. Then total number of parallelograms so formed is \[^{m}{{C}_{2}}{{\times }^{n}}{{C}_{2}}\,\,i.e.,\frac{mn(m-1)(n-1)}{4}\].

(5) Given \[n\] points on the circumference of a circle, then

(i) Number of straight lines \[{{=}^{n}}{{C}_{2}}\]   

(ii) Number of triangles  \[{{=}^{n}}{{C}_{3}}\]

(iii) Number of quadrilaterals \[{{=}^{n}}{{C}_{4}}\].

(6) If n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. Then the number of part into which these lines divide the plane is \[=1+\Sigma n\].

(7) Number of rectangles of any size in a square of \[n\times n\] is \[\sum\limits_{r=1}^{n}{{{r}^{3}}}\] and number of squares of any size is \[\sum\limits_{r=1}^{n}{{{r}^{2}}}\].

(8) In a rectangle of \[n\times p\,\,(n<p)\] number of rectangles of any size is \[\frac{np}{4}(n+1)\,(p+1)\] and number of squares of any size is \[\sum\limits_{r=1}^{n}{(n+1-r)\,(p+1-r)}\].