The sum of all absolute values of the differences of the numbers $1, 2, 3, \ldots, n$,taken two at a time,i.e.,$\sum \limits_{1 \leq j < i \leq n} |i-j|$ equals:

There are $n$ points in a plane,of which $p$ points are collinear. How many lines can be formed from these points?

If the number of diagonals of a regular polygon is $35$,then the number of sides of the polygon is

In a plane,there are $37$ straight lines,of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides,no three lines pass through one point,no line passes through both points $A$ and $B$,and no two lines are parallel. The number of intersection points the lines have is equal to:

Some couples participated in a mixed doubles badminton tournament. If the number of matches played,such that no couple played in a match,is $840$,then the total number of persons who participated in the tournament is $........$.

If three vertices are chosen from the six vertices of a regular hexagon and joined to form a triangle,what is the probability that the triangle formed is an equilateral triangle?