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

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

If three distinct vertices are chosen at random from the vertices of a cube,then the probability that they form the vertices of an equilateral triangle is:

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track. Further,each pair of nearest stations is connected by a blue line,whereas all remaining pairs of stations are connected by a red line. If the number of red lines is $99$ times the number of blue lines,then the value of $n$ is:

The number of diagonals in a polygon is $20$. The number of sides of the polygon is:

If a polygon has $44$ diagonals,then the number of its sides is: