SOME IMPORT ANT RESULTS ABOUT POINTS :
If there are n points in a plane of which m (< n) are collinear, then
`(a)` Total number of different straight lines obtained by joining these n points is `text()^nC_2-text()^mC_2+1`
`(b)` Total number of different triangles formed by joining these n points is `text()^nC_3-text()^mC_3`
`(c)` Number of diagonals in polygon of `n` sides is `text()^nC_2-n= (n(n-3))/2`
`(d)` lf m parallel lines in a plane are intersected by a family of other n parallel lines. Then total number of parallelograms so formed is
` text()^nC_2 xxtext()^mC_2=(mn(m-1)(n-1))/4`
`(e)` If there are n points in a plane out of these points no any three are
collinear, then
(i) Total points of intersection of the lines joining these n points `= text( )^PC_2`
,where `p = text( )^nC_2`
(ii) If n points are the vertices of a polygon, then total number of
diagonals `= text( )^nC_n-n = (n(n-3))/2`
`(f)` n straight lines are drawn in a plane such that no two of them are
parallel and no three of them are concurrent. Then, number of parts into
which these lines divides the plane is equal to
`1+ sum_(k=1)^n k i.e., (n^2 +n +2)/2`
`(g)` If m parallel lines in a plane are intersected by a family of other n
parallel lines. Then, total number of parallelograms so formed
`= text( )^mC_2.text( )^nC_2 , (mn(m-1)(n-1))/4`
`(h)` `text(Number of Rectangles and Squares)`
(i) Number of rectangles of any size in a square of `n xx n` is `sum_(r=1)^n r^3` and
number of squares of any size is `sum_(r=1)^nr^3`
(ii) In a rectangle of n x p (n < p) number of rectangles of any size :is
`(np)/4 (n + 1) ( p + 1)` and number of squares of any size is
`sum_(r=1)^n(n+1-r)(p+1-r)`
`(i)` If there are n rows, first row has `alpha_1` squares, 2nd row has a 2 squares, 3rd
row has `alpha_3` squares, ... and nth row has O'.n squares. If we have to filled up
the squares with `beta X_s` such that each row has atleast one X.
The number of ways = Coefficient of `x^(beta)` in
`(text( )^(alpha_1)C_1x + text( )^(alpha_2)C_2x^2 + .......... + text( )^(alpha_1)C_(alpha_1)x^(alpha_1)) xx (text( )^(alpha_2)C_1x + text( )^(alpha_2)C_2x^2 + .......... + text( )^(alpha_2)C_(alpha_2)x^(alpha_2)) xx ............xx (text( )^(alpha_n)C_1x + text( )^(alpha_n)C_2x^2 + .......... + text( )^(alpha_n)C_(alpha_n)x^(alpha_n))`
`(j)` Number of triangles formed by joining vertices of convex polygon of `n` sides is `text()^mC_3` of which
`(i)` Number of triangles having exactly two sides common to the polygon `= n`
`(ii)` Number of triangles having exactly one side common to the polygon`= n(n - 4)`
`(iii)` Number of triangles having no side common to the polygon`=(n(n - 4)(n - 5))/6`
SOME IMPORT ANT RESULTS ABOUT POINTS :
If there are n points in a plane of which m (< n) are collinear, then
`(a)` Total number of different straight lines obtained by joining these n points is `text()^nC_2-text()^mC_2+1`
`(b)` Total number of different triangles formed by joining these n points is `text()^nC_3-text()^mC_3`
`(c)` Number of diagonals in polygon of `n` sides is `text()^nC_2-n= (n(n-3))/2`
`(d)` lf m parallel lines in a plane are intersected by a family of other n parallel lines. Then total number of parallelograms so formed is
` text()^nC_2 xxtext()^mC_2=(mn(m-1)(n-1))/4`
`(e)` If there are n points in a plane out of these points no any three are
collinear, then
(i) Total points of intersection of the lines joining these n points `= text( )^PC_2`
,where `p = text( )^nC_2`
(ii) If n points are the vertices of a polygon, then total number of
diagonals `= text( )^nC_n-n = (n(n-3))/2`
`(f)` n straight lines are drawn in a plane such that no two of them are
parallel and no three of them are concurrent. Then, number of parts into
which these lines divides the plane is equal to
`1+ sum_(k=1)^n k i.e., (n^2 +n +2)/2`
`(g)` If m parallel lines in a plane are intersected by a family of other n
parallel lines. Then, total number of parallelograms so formed
`= text( )^mC_2.text( )^nC_2 , (mn(m-1)(n-1))/4`
`(h)` `text(Number of Rectangles and Squares)`
(i) Number of rectangles of any size in a square of `n xx n` is `sum_(r=1)^n r^3` and
number of squares of any size is `sum_(r=1)^nr^3`
(ii) In a rectangle of n x p (n < p) number of rectangles of any size :is
`(np)/4 (n + 1) ( p + 1)` and number of squares of any size is
`sum_(r=1)^n(n+1-r)(p+1-r)`
`(i)` If there are n rows, first row has `alpha_1` squares, 2nd row has a 2 squares, 3rd
row has `alpha_3` squares, ... and nth row has O'.n squares. If we have to filled up
the squares with `beta X_s` such that each row has atleast one X.
The number of ways = Coefficient of `x^(beta)` in
`(text( )^(alpha_1)C_1x + text( )^(alpha_2)C_2x^2 + .......... + text( )^(alpha_1)C_(alpha_1)x^(alpha_1)) xx (text( )^(alpha_2)C_1x + text( )^(alpha_2)C_2x^2 + .......... + text( )^(alpha_2)C_(alpha_2)x^(alpha_2)) xx ............xx (text( )^(alpha_n)C_1x + text( )^(alpha_n)C_2x^2 + .......... + text( )^(alpha_n)C_(alpha_n)x^(alpha_n))`
`(j)` Number of triangles formed by joining vertices of convex polygon of `n` sides is `text()^mC_3` of which
`(i)` Number of triangles having exactly two sides common to the polygon `= n`
`(ii)` Number of triangles having exactly one side common to the polygon`= n(n - 4)`
`(iii)` Number of triangles having no side common to the polygon`=(n(n - 4)(n - 5))/6`